What is too big of current that will fry the voltage regulator?

Those figures assume infinitely large heatsinks.

At this point, a lot of people will suggest a dc-dc converter or else say, "the pcb acts like a
heatsink", but of course it's not inifinite, and no one has probably tried to measure it's
effectiveness in any event.

Reality is quite different.

So, what is obviously clear is that the figures I gave are a 'best' case estimate, and in actual
practice, the v.reg will run even hotter yet. I personally don't like having my pcbs getting
so warm as 176 degF.