With a locked screen, key presses go to the password field. I have twice caused my user account to become disabled due to too many password attempts while cleaning my keyboard.

A pro tip from a Mac sysadmin who gets to clean a lot of filthy laptops: https://folivora.ai/keyboardcleantool

What's the equivalent of "locking the screen" for cleaning an Apple TV's paired remote?

Tell the Apple TV to restart and you have 15 seconds to clean the remote.