I think that unfortunately, the Democrats will have to cave in the end. Its hard to win a game of chicken when the other driver is a member of a death cult.
The real difficulty is that the Republicans have different motivations that the Republicans. The Democrats have to be concerned about the perception of the buisiness community and the public at large, and also might have a slight interest of not having the country go up in flames. All the Republicans care about is their base, and that base sees signs of weakness and compromise in the face of the Democrats as the most cardinal sin. They have already demonstrated that they are willing to die in order own the libs, so destroying the nations credit rating is no big deal.
They way it will probably go down is that there will be a govenment shutdown for a couple of weeks. This may be less disruptive than normal. We’ve been through this before, everyone knows the drill. Also due to the pandemic many government workers are already staying home, which makes it easier for some of them (myself included, don’t tell anyone) to cheat and do a little bit of illegal clandstine government work on the side to keep the backlog from being too overwhelming when we get back to work.
The Govenment will creep closer to default there will be multiple cloture votes against default in the senate with Republicans repeatedly voting no. But on the last day the Dems will cave and pass it through reconcilliation to avoid default. The Republicans will loudly claim victory and try to spin this that it proves that the crisis was the Democrats fault since they could have stopped it at any time without the Republicans help. The Dems will argue that they can hardly be blamed for obstructing a bill that they uniformly voted for and the Republicans uniformly voted against. Partisans on either side will accept their sides version of events while the small sliver in the middle will take the easy way out by saying that both sides are equally to blame.
Hopefully by waiting until default became big news, the Democrats will neutralize the strategy of claiming that voting against the debt ceiling indicates financial responsibility, but one can never underestimate the stupidity of the American public.