I debated putting an answer at all because I don't have time to hunt down quotes, but since there seems to be an utter lack of a key perspective I suppose I'll have to write something.
So far all links tried to look at what the party leaders and strategists thought of as reasons for this disparity, which is a valid perspective. Though to a degree I think at high level strategy perspective republicans have mostly written off minorities as not a demographic they are likely to win. Democrats already have a strong sway over this demographic, trying to draw them away would require significant expenditure of effort or change of platform that may in turn alienate some of their existing voter base. It's simply not cost effective to dedicate the effort it would take to claim back a large perspective of this voter base. Sure if there is any low effort things that can be to make the party slightly more appealing it may be considered, but it's not cost effective to dedicate a significant effort to this cause. To give a sample when Mccain was running for president and republicans felt they needed to find a new source of voters to draw upon they explicitly set their sights on low income and/or uneducated white voters instead of minorities because that was a voter base they thought they could more effectively appeal to.
In this regard from a high level strategic perspective there is no real surprise by analysists that minorities vote democratic. Democrat's are known as the party for minorities, they already have had a strong ownership of this demographic for a long time, and considering that it's not a demographic considered worth targeting for republicans while Democrat's are actively targeting it it makes sense the majority would end up with the party actively appealing to them.
However, there is a second perspective, what does the average republican voter feel is the reason for this discrepancy? I'm not talking about the strategist and high level leaders, I mean the average John Doe republican voter think? I couldn't find a good study on this with quick googling sadly. However, I do have a decent proxy for this perspective, Fox News.
69% of republican voters say they Fox News a credible source, which is a significantly higher percentage then any other news source in the study. It was the single most cited source of news by republicans and 93% of those who got their news from fox were republican voters. I think Fox News represents a key source of information and thus source for explanations for a non trivial percentage of republican voters. Thus to figure out what they average voter thinks we can go to Fox News and see what their explanation is. This of course won't cover what every republican voter believes, but it will suggest the explanation likely to be believed by a non trivial percentage of them.
So how does Fox explain the discrepancy? Note for the rest of this answer I'm speaking from a Fox republican point of view. I will fully admit to the occasional resorting to exaggeration in my language choice to make a point. While the main talking points seem to persist across all Fox shows how exaggerated vs nuanced the actual talking points are depends heavily on which Fox personality you're listening to. Still it's both easier to make my points clearly and more fun to explain things using the more exaggerated voice so that's what I'm going with :P
Minorities will vote for whoever gives them the most handouts
In a way this is not really much of a surprise. If you look at any voting demographic it's generally a true statement that they are going to vote for whatever party the think will best benefit them; though it's important to point out that what matters here is what the voter thinks is best for them, not rather that belief is actually true.
What does matter here is how this basic principal is portrayed. There is a certain degree of blame put here, as if minorities are wrong for voting in their best interest instead instead of what is 'best for America', despite the fact that practically every voting demographic will vote primarily based off of what they believe most benefits them. The OP's already referenced claim of possible racist subtext can apply here, I've definitely heard claims that there is some dog whistling going on and attempt to sell blaming economic woes to those who already hate minorities. However, even ignoring claims of racism an argument for blaming minorities for voting in their best interest can still be made if it's believed that vote is excessively harmful to the rest of America's interest, as we'll get to below.
Regardless of the reason for the blame it does play a role in a subtler sort of argument, that it's okay not to have minorities voting for Republicans since they are guilty of 'bad' voting and thus not the sort of voters a good Republican party would want to court.
I put this first since it influences the rest of the issues listed below, though I don't consider it alone a major contributor for how republican voters explain the discrepancy.
Democrats are targeting minorities to the expense of the rest of the nation
Here is our smoking gun, the single biggest and most used argument, by a fair margin, to explain the voting discrepancy.
The basic argument is that democrat's are dependent on minority voters so much that they will do anything to appeal to them, even if that thing is harmful to the country as a whole. Democrats will bankrupt the country, tax everyone to death, run up the national deficit, and basically screw over all non-minorities in whatever manner it takes to give handouts to minorities to buy their votes. After all democrat's don't really care about America, they just care about winning votes at all cost.
Using this argument it makes total sense that minorities would vote for the people giving them handouts. Though going back to the first argument this is the other major explanation for why minorities are blamed for voting for the party that benefits them the most; because it's encouraging the democrat's continued abuse of the nation to give them handouts and is thus unethical.
Of course the problem with the Democrat's approach is that it's unsustainable and is going to destroy America's economy. By contrast Republicans don't get many minority votes because they refuse to tank our economy just to give out handouts. It's not that they are doing something wrong to alienate minorities, they simply refuse to buy votes at the expense of the average tax payer the way Democrats do!
Minority voter, what minority voters?
There is always the age old and simple strategy of just not mentioning your deficits. Don't bring up the difference, focus on the demographics they have strong voter turn out on, and redirect anyone trying to ask why minority voters are underrepresented.
This general strategy is by no means limited to this one issue, or Republicans alone. Every group out there will try to redirect people from, and avoid talking about, their weaknesses. However it still seems worth pointing out. The majority of republican voters aren't asking why there is a difference in voter turn out for minority voters in the first place. It's not a topic discussed in Republican leaning media and if you just avoid the topic most won't think to ask about it. I mean there is a reason every group out there uses the tactic of trying to avoid talking about their weaknesses, because it works!
Democrats and the liberal media keeps unfairly trying to frame us as bigots to drive away our voters
You said yourself every explanation you found for the discrepancy seemed to come down to accusing Republicans of racism. That isn't because the republicans are racist, but because the liberal media is trying to smear the republican party. If Democrats can convince everyone that Republicans are all racist then of course minorities wouldn't want to vote for them. Thus both Democrats and the Democratically controlled liberal media tries to find any excuse they can to play the race card and demonize Republicans just to drive minority voters away from them.
There is no real racism in the Republican party of course, but so long as democrat's can make them seem that way it still prevents them from getting minority votes.
We totally have minority voters, look right here is my black best friend!
Okay this is at most a minor subcurrent rather then a primary accusation. However, Republicans will do their best to hold up examples of black, or other minority, voters and leaders within their party to further show they are a diverse party. This usually goes hand in hand with claiming the media is exaggerating the discrepancy between minority voter turnout and implying this isn't as big a problem as the liberal media would have you believe.
Democrats are unethically encouraging (illegal?) immigrant voters to bring up their numbers
The Democrats want weak boarder crossing, lenience or even a path of citizenship for illegal immigrants, easy asylum, sanctuary cities, and assorted similar policies all for the goal to get more (illegal?) immigrants to vote for them.
This argument can range from relatively reasonable accusations that the democrats favor lenient immigration laws partially because those minority immigrants are more likely to vote for them to claiming they are benefiting from illegal immigrant votes as well to the extreme fear mongering conspiracy theories of intentionally bussing in illegal immigrants to voting stations to get more illegal votes for democrats.
At the most reasonable side of the argument it's undeniable that democrats both tend to favor easier immigration and tend to benefit from higher turnout from immigration votes. Now how much the motivation for the former is driven by the latter benefit is hard to say, and likely varies depending on which democratic your talking about, but at the very least it's not hard to imagine this being a non-trivial motivation for lax immigration law.
Of course the most reasonable argument might seem irrelevant to OP question, it's only stating democrat's benefit from immigrants, not explaining why minority immigrants would vote for them. However, it still can work as sort of guilt by association argument. Ie democrats are wrong for favoring unregulated immigration just to get more votes, (minority) immigrants are benefiting from this wrong, therefore (minority) immigrants are wrong for voting and thus we don't want those filthy minority immigrants voting for us anyways.
On the more extreme side of things is varying arguments that Democrats are benefiting from illegal aliens voting, and may be actively trying to encourage it in more extreme claims. This version basically argues that the only reason that Democrats seem to have such a lead in minority votes is because a non-trivial number of those minority votes are illegal votes that shouldn't have counted. If we just stopped all that illegal voting, say with a boarder wall to keep them out, demolishing sanctuary cities, and voter ID to make sure only legitimate voters vote, then the lead in minority voters Democrats seem to own will fade since it's no longer being fueled by all those illegal votes.
You might argue that there is little evidence of illegal voting, or that even if illegal's were voting they likely would not be included in the polls that show democrats being favored by minorities. However what matters is what one believes, and there is plenty of studies and evidence that a non trivial percentage of Republicans believe illegal voting has happened and it's messing up demographic data. We had a little riot driven by that belief you might have heard of after all ;).
So the TL/DR version of this answer is that Republicans accuse Democrat's of using immoral, unsustainable, methods to court minority voters and that since the Republicans are unwilling to resort to such tactics it's understandable they wouldn't be able to compete with those that did.