Debt default endgame?

On November 1,  $25 billion in Social Security checks are due to go out.  Some experts anticipate that this will be the forcing event for a debt default resolution, if there hasn't already been a stock market crash or other dire macroeconomic event.

$18 billion in Medicare payments are also due to go out on November 1.  But, as those go to providers, rather than directly to seniors, the short-term optics aren't quite as politically intense.

In the scenario that I've been reading about, the Administration announces, a few days prior, that Social Security checks won't be going out due to insufficient funds.  It attributes this to the House's unwillingness to raise the debt ceiling, no doubt with a personial reference to the Speaker in particular.

In an "optimistic" version of the scenario (keeping in mind that this is a relative term), the House then folds.  But does this necessarily follow?  The House Republicans might instead respond by saying it's not their fault but the Administration's, both for not "negotiating" and for not prioritizing these payments over all the others that the government owes.  They might even vote budgetary authorization for the Treasury to issue and use new debt just for Social Security payments, a la their "partial shutdown" efforts over the last few days.

We might then face a situation where millions of seniors are screaming (and suffering) because they haven't gotten their checks, and yet the battle continues.  Far-fetched, perhaps, but I don't think entirely impossible so long as blame can be contested.  Once you start playing chicken it's hard to fold, and the Republicans might feel they'd simply gone too far to quit the game with nothing.