No one is claiming that this kind of argument constitutes a proof, but this kind of back of the envelope calculation is very powerful. For example in the theory of prime numbers there is a heuristic that says that primes roughly behave like randomly selected numbers where Prob(N is prime) = 1 / log(N) [this is a simplification, but that's the crux of it]. With this heuristic you can accurately predict whether a large class of statements about primes are true or false, and can get extremely precise estimates about things like "how many twin primes are there less than N", or "how many solutions in primes are there to p1 + p2 = 2*N, for some huge N", which is one way of phrasing two famous prime number conjectures.
It doesn't lead you directly proof - but often just knowing what the answer 'should be' can be a real guiding light.
It doesn't lead you directly proof - but often just knowing what the answer 'should be' can be a real guiding light.