It’s memory holed now but the reason Obama was able to eek out a re-election victory in 2012 despite the debt ceiling debacle a year prior was because neither Mitt Romney (“47 percent”, anyone?) nor Todd Akin could keep their mouths shut (Akin’s gaffe was also the reason Claire McCaskill’s goose was saved; she was losing in the polls before those comments, and even then there was zero reason to believe she was going to get a third term after that given how both Romney and Trump swept Missouri by landslides).
Given how Biden is nowhere near as popular among the Democratic base as Obama was, I don’t think any GOP gaffes will save him once this debt ceiling debacle is over, default or no default.