Do removable shower hair catchers stay put without slowing the drain?

A removable hair catcher seems handy, but I’m not sure how well it works with different shower drain covers. Some drains have a cover or strainer that’s meant to stay in place, and I’d be wary of setting something on top if it could rock, shift, or interfere with a stopper.

What drain-cover styles tend to work with a separate catcher, and how can you tell it’s seated securely without blocking anything? If drainage starts slowing, is clearing the catcher usually the first maintenance step, or can it be a sign the catcher doesn’t fit the drain well?