I can't vouch for the procedural accuracy of these claims, but the problem seems to be that the procedure would take
around 10 days, and the US at best has 26 days before it defaults (according to that source). Other sources say Oct 15 might be the day of default, so that would give less room.
There's a procedural dispute whether the thing can be done without going through a Senate committee, some obscure stuff about "section 304" and the Senate parliamentarian saying it must go through a committee. And the Senate committee might supposedly block it if no Republican shows up at all at the session. Apparently that block can be overridden, but the whole affair could take 10 days or so.
Here's a slightly more intelligible write-up, but alas pretty long:
Democrats, in particular House Budget Committee Chair John Yarmuth, have argued that there isn’t sufficient time to use the reconciliation process to raise the debt ceiling. This isn’t strictly true—if they really hustle, they can perhaps complete the process in as little as 10 days. [...]
Democrats are using the budget reconciliation process to pass their big bill, including a laundry list of priorities. This requires first passing a budget resolution to lay out instructions for committees to craft the reconciliation bill, which House and Senate Democrats did in August. As we know, they did not include a debt ceiling hike in their resolution. [...]
The Senate parliamentarian, the top expert on Senate procedural matters, who acts as a referee advising what the chamber can and cannot do under the rules, issued an opinion in June setting ground rules for revising a previously adopted budget, as authorized by Section 304 of the Congressional Budget Act passed in 1974. [...]
So here’s what the Senate can do: Revise the existing budget resolution for fiscal year 2022, which was used to set instructions to craft the reconciliation bill. The revised resolution can ask the Senate Finance Committee and the House Ways and Means Committee to raise the debt limit to some dollar amount. (The parliamentarian would likely not accept a suspension of the debt ceiling, meaning that they will have to raise it.)
The initial fiscal year 2022 budget resolution, which passed in August, did not have to go through a Budget Committee markup, but was auto-discharged. The Budget Act requires that the Budget Committee report a budget resolution by April 1. If the committee is late to report a budget resolution, it is automatically discharged from consideration by the committee, meaning that it can just go straight to the Senate floor.
Parliamentarian Elizabeth MacDonough said in her June opinion that a budget resolution revised under Section 304 would not be eligible for an auto-discharge. This means that the revised resolution will have to be considered by the Budget Committee before it can make it to the Senate floor.
If they want to, Republicans can gum up the works by refusing to attend the markup and denying the committee a quorum. But assuming they do attend, the committee will likely deadlock, as every committee has an equal party split. Under current Senate rules, the bill would still be discharged to the floor if there is a tie vote in the committee, but the discharge procedure is complicated (shocker) and eats up valuable time.
And that's apparently not all of it.
There could yet be some further drama with Washington’s favorite fixture, the Senate parliamentarian. Given the rarity of using Section 304 to revise a budget resolution, MacDonough may have to rule on how the process should work. [...]
Or—even worse for Democrats—she could issue an opinion saying that Democrats’ big priorities can’t be addressed until fiscal year 2023. This could mean waiting until next April to take up the bill again, and by that point, the campaigns for the midterm elections will be in full swing. It’s a lot harder to get stuff done in an election year, especially massive stuff that could reshape the country.
[...] this could be a ploy by Senate Minority Leader Mitch McConnell to block the larger reconciliation bill. His office could argue to the parliamentarian that revising the resolution should wipe out their existing bill or that it should be pushed back.
MacDonough ruled earlier this year that a federal minimum wage increase could not be included in the American Rescue Plan, the coronavirus relief measure passed by Democrats using reconciliation in March. More recently, she issued an opinion that, under Senate rules, immigration reform could not be included in the reconciliation bill Democrats are currently crafting.
Theoretically, the parliamentarian could be ignored—she advises the presiding officer of the Senate, the vice president. But Vice President Kamala Harris is unlikely to take that option, which has only been done once before, by Vice President Nelson Rockefeller in 1975. MacDonough could be overruled by a simple majority vote, but that would require support from all Democrats, which is, again, unlikely.
Basically, it might come down to exercising a "nuclear option" either in voting to overrule the Parliamentarian or (perhaps even more controversially) choosing to ignore her. According to Wikipedia, the 1975 incident essentially broiled down to changing the rules immediately thereafter:
That ruling was extremely controversial, to such an extent that the leaders of both parties immediately met and agreed that they did not want this precedent to stand, so the next week the Senate altered the rule under consideration via standard procedure.
If you actually intended to ask the slightly different question why the Democrats haven't included the ceiling raise in the August reconciliation that's almost certainly for "optics" reasons as they were hoping to get the Republicans on board on the ceiling issue but probably have no hope of them voting for the $3.5T spending etc.
Pelosi said yesterday that raising the celling has "always been bipartisan". (It could be an interesting fact check if that is so, but for the purpose of answering your question what Democrats say about this, it's enough that she said that.) In the same video segment McConnel says that Republicans were "left to do it alone in the early 2000s".