Apologies to rain on the parade, but this sort of argument is very well known and commonly used in number theory. Since these techniques don't lead to proofs, they may not get as much attention in standard math education, but they are still well-known. Many number theory papers use this technique (often with more refinements) to estimate the number of expected solutions. I would be stunned if it wasn't used on Fermat's last theorem long before Feynman did.

For an exposition on some heuristics mathematicians use, https://terrytao.wordpress.com/2015/01/04/254a-supplement-4-... is a good read (although it's aimed at readers already having some background in number theory).