Should one avoid lots of use statements in a custom modules .module file?
The use statements alone are definitely not going to be a performance concern. I doubt that we're even talking about nanoseconds here. So that really doesn't matter. I'm guessing that this is resolved at compile time anyway.

Especially in a .module file, use statements usually grow, the more code you have, the more use statements, as a .module file is often doing many different things, implementing various hooks and so on.

If you want to optimize something, look at your actual code, forget about the use statements. Some hooks can be moved into include files, especially those that are only needed when rebuilding caches, like hook_views_data().  Other things can be done in services. But also here, the argument is most likely not going to be runtime performance, parsing code is really, really fast. It's your performance in understand the code, finding bugs, writing  test and so on. Optimize for that.