It's probably not exactly going to be hot, but even failing execve is inherently semi-expensive since it needs to be a syscall and incurs context switches.

It's just outweighed a couple orders of magnitude by all the overhead that comes with a successfully launching another executable unless you have, like, a thousand junk paths in your PATH.

Theoretically can be. Every command you invoke without a path will need to look up PATH.

In practice well behaving shells cache the contents of PATH to speed up operations.