Can somebody heighten this contradiction?

It’s debt ceiling time again, although the real crisis won’t roll around until June or so.
This farce is going to continue to play out whenever the GOP controls the House but not the other two branches of government, so in other words often.
Someone explain to me why this isn’t a good approach: Biden tells the Kevin McCarthy that he’s not going to negotiate on this, period. It’s terroristic hostage-taking and America Does Not Negotiate With Terrorists. So go ahead and default, and by the way it’s illegal for the government to issue any Social Security checks or Medicare reimbursements if the government is in default (I don’t care whether this is strictly technically true or not. It’s close enough to true for practical political purposes).
Then one of two things happens: McCarthy blinks, or the government defaults, and tens of millions of people don’t get their SS checks for a month, the Republican party is — rightly — blamed for this, and suffers an electoral catastrophe as a result. Yes I KNOW the liberal media ™ will BothSides it, while Fox News et. al. will explode in an apoplectic frenzy. Doesn’t matter, won’t work.
Obviously defaulting would be really bad in the short term. But you have to bite the bullet on this issue sometime, so why not now, when the GOP House majority is the smallest it’s ever going to be in regard to this matter?