There were extensions that got around this, though. iTunes for the classic Mac OS (and I'm pretty sure SoundJam before it) could continue to play music with a menu open, for example.
But those tended to have some pretty gnarly limitations (like I think in interrupts you can't allocate memory) so AFAIK they were only used for stuff like real-time audio, I dunno if anyone ever used those to do screen drawing, so in practice I can't think of anything that would interfere with menu drawing.