Why won't the Republicans use a superdelegate system like the DNC in their nomination process? Is there a reason why they didn't push for this yet? What does the Republican party think of a superdelegate system and why did the DNC opt for it while the Republican party has not yet implemented such a system?
A better way to look at this question is to ask why Democrats do have such an expansive superdelegate system (Republicans kiiiiiiiiind of have superdelegates, but far less and bound to the results of state primaries).
In 1968, the Democratic Party's primary system was exposed as more or less a farce. Due to a lot of unforeseen circumstances-- such as sitting president Lyndon B. Johnson deciding not to seek reelection mid-run, and the assassination of Robert Kennedy-- Vice President Hubert Humphrey didn't receive a single vote in a state primary, but ended up winning the nomination due to a bit of wheeling and dealing behind the scenes. Particularly galling to most Democratic primary voters was that Humphrey was pro-Vietnam War, despite the fact that the vast majority of actual votes had gone to anti-war candidates like Kennedy and McCarthy. All the while you have riots and violence outside the convention, and the whole thing is a PR disaster.
So following the election, the party reforms the party primary process to shift the power to actual Democratic voters. But what immediately follows is a period of Republican presidential election domination. George McGovern, at the time probably the most liberal candidate to date, gets flattened by by Nixon. Jimmy Carter (not exactly in the liberal branch of his party, but certainly a populist) manages to beat a Watergate-tainted and unelected Gerald Ford, but gets flattened by Reagan four years later. Democrats run liberal Walter Mondale four years later, and the result is the most lopsided presidential election to date with Reagan winning 49 states.
So now Democrats are looking at a situation where the only successful presidential candidates in their lifetime were chosen in smoked-filled rooms, with the exception of 1976, when the Republicans were so badly hurt by Watergate that they could've run a ham sandwich. They come to the conclusion that perhaps they've gone a bit too far; perhaps there can be a system where the voters have most of the votes, but the party elites can still act as a check. The 1984 election is the first Democratic election with superdelegates, and they've stuck ever since. (Not that it helped. Again, 49 states.)
So why have Republicans never followed suit? Bluntly, they've never needed to. Superdelegates first and foremost are designed to check the ideologically entrenched wing of your party. The GOP has often seen the right-wing candidate beat the more centrist candidates for the nomination: Reagan over H.W. Bush, W. Bush over McCain, Trump over a lot of people. But what do those candidates have in common? They won.
That's your simple answer. Democrats had a series of serious electoral setbacks that, rightly or wrongly, they viewed as requiring stricter party control of the nomination process. Republicans never really had one.
Republicans do have a superdelegate system, except that only 7% of the Republican nominating delegation. The current system makes superdelegates nominate the primary candidate who won their state and yes, there is a lot of talk about changing the Republican Nomination System. This all started when Beau Correll Jr., an attorney from Virginia filed a lawsuit because he was required to vote for Trump because Trump won Virginia in which he believed that violated his 1st Amendment right.
The direct roots of the Democratic superdelegate traces to 1972, and George McGovern.
True that 1968 was a mess, plus a rather dull Humphrey beating out the only remaining populist choice, Gene 'clean for Gene' McCarthy, but what really did them in that year was the protests at the 1968 Democratic convention in Chicago, and especially Daley's very heavy handed response to protesters. The Democrats were tarred and feathered with images of Chicago police not just arresting but severely beating protestors.
Those were very powerful images and not good ones... I watched that in 1968 when I was 7 years old, and I can still see the nightsticks going up and down, up and down.
If 1960 was the year that television showed how it could promote a photogenic and congenial candidate, 1968 showed how television coverage could wreck a campaign with a single bad event.
1972 was the election that directly prompted the superdelegates.
It wasn't that a left leaning ideologue was nominated, but that McGovern ran probably the most incompetent and incoherent campaign... ever. Just about everything McGovern did in that campaign was a disaster, starting with his choice for VP, Thomas Eagleton. After Eagleton had been accepted, it came out that he had been through electroshock therapy treatment for mental issues... a fairly drastic treatment.
It's important to remember that McGovern didn't fail due to his ideology, he failed due to his utter incompetence in the mundane matters of running a campaign. He stuck with Eagleton far longer than he should have, and his ability to organize and drive a campaign was just about nil. Presidential campaigns are a test of the candidate's ability to not just have a vision, but have the skills to make that happen. McGovern failed that test badly.
Despite the triple bonus of the anti-war sentiment, hints of the Watergate break-in made public before the election, and hard right George Wallace running an independent campaign that siphoned off Republican votes, McGovern was still roundly defeated, lost 49 out of 50 states.
Thus, the superdelegates, to give the party more of a voice in the selection of a presidential candidate... to prevent a McGovern debacle from ever happening again. Not so much to control ideology, but to keep a person who can't handle the job from getting the nomination... again. In theory, there are enough to sway a close race, but not enough to dictate the choice in contravention to the majority of Democratic voters. In practice, they give the party more of a 'kingmaker' ability... great influence with candidates, and enough to where they can keep an incompetent populist from damaging the party.
Losing a presidential election as badly as McGovern did can damage the party for years. Had it not been for Watergate, his defeat would probably have haunted the party as badly as Carter's extreme loss did in 1982.
There is one downside to the concept of superdelegates: they can lead to voter disenfranchisement, as in the party doesn't care about what I think. That may have been evident in 2016, when superdelegates kept the rising Sanders campaign from ever having a chance to actually win. That, plus the party appearing to be actively undermining the Sanders campaign in the primaries, may have led to a number of ardent Sanders supporters simply not voting in November, because the party didn't listen to them. As the DNC reduced the number of superdelegates after 2016, they may be acknowledging this side effect.
The Republicans don't have nearly the level of superdelegates that the Democrats do. But, if one remembers who the party chose as the anointed candidate in 2016... the rather dull Jeb Bush with his association with W... perhaps that worked out for the best for them.