There might just be a terminological distinction here. When I think of the reasoning used by mathematicians/physicists, I think of the reasoning used to guess what is true - in particular to produce a theory with >50% confidence. I don't think as much of the reasoning used to get you from >50% to >99%, because this is relatively superfluous for a mathematician's utility function - at best, it doubles your efficiency in proving theorems. Whereas you are concerned more with getting >99%.

This is sort of a stupid point but Euler's argument does not have very many parts, and the parts themselves are relatively strong. Note that if you take away the first, conceptual point, the argument is not very convincing at all - although this depends on how much calculation of how many even zeta values Euler does. It's still a pretty far cry from the arguments frequently used in the human world.

Finally, while I can see why Euler's reasoning may be representative of the sort of reasoning that physicists use, I would like to see more evidence that it is representative. If all you have is the advice of this chauffer, that's perfectly alright and I will go do something else.