I think this is somewhat orthogonal.
Unexpected behaviour and errors can be caught by means other than constant recompilation, and tools exists to do just that. In IDEs, these tools tend to become invisible, seamless, and can be relied upon without much configuration; for those accustomed to less integrated environments, the manual configuration required to get these tools up and running becomes a hassle.
I suppose using those tools efficiently and constant recompiling are different ways to solve the same problem, but I wouldn't say it's solely the compiler's job even in the latter case. For languages that aren't super-fast for compiling like Rust, Swift, etc. there may be much more reliance on those other tools.
There’s two sides to this: on the one hand, you don’t have to keep as much info in your head because the compiler will tell you if you ask it. On the other hand, when you do try to keep state in your head, the compiler is able to double-check that for you so that you don’t have to do it perfectly.
With a language like Python, you’re working without a safety net, so you make small, cautious, steps in order to not fall. A strongly-typed language like Rust enables you to make larger, bolder, changes in one go, confident that the little mistakes will get caught.
Doesn't that partially defeat the point of having a powerful compiler? The more work the compiler does, the less we have to keep track of in our head (and conversely, much of the pain of coding in a language like Python is the amount of stuff we need to mentally keep track of).