"Extrapolation of the expansion of the Universe backwards in time using general relativity yields an infinite density and temperature at a finite time in the past."Works out fine working backwards. It's not-infinite in the moment before it gets infinite, and being compressed. Well, works out enough, I suppose.
The problem is going forward. How does the infinite state know it is supposed to become not-infinite? How does it run out of energy? The past doesn't know the future. Mathematically, infinities can't know how big they are. Technically speaking, math never deals with infinities per se, but with limits. Problem is, an infinite physical state can't use l'Hopital's rule on itself. It doesn't know what function it is the limit of. Which means it's not the limit of a function, but a real, honest-to-goodness infinity.
Or, put another way, when your model assumes the infinity is the limit of some function, I can counter with an identical particle that's the limit of some other function, which is the classic test of 'undefined' in mathematics.
Which means it can't run out. An infinite state should stay infinite forever.
Or, it is strictly speaking undefined.
I should stress I have no real issue with the Bang in general terms. However, specificity is critical to physics. Fixing this infinite is likely to have wide-ranging consequences, not at all subtle.
Hopefully a theme is becoming clear: physicists do not think about their theories anymore, and it is causing serious issues. They do the math, trust they've done the right math, and haven't.