Ribbon Tweeters and Animal Hair

Cats will use the grill cloth as a scratching post. They will not bother the drivers. You can get electrified plastic mats that you can put around the speaker. Any biologic organism that steps on the mats will get an uncomfortable buzz just bad enough to relate to that organism that this is a no go zone. You keep the mats there for 6 months and you can remove them.
If the speakers are cool looking with the grills off then keep them off. Pet hair is heavier than air. As long as the pet is not above the speaker hair is not an issue. Ribbons can be very delicate, you never want to vacuum over them or blow into them. You want to keep a spare ribbon in hand in case something happens. You wipe the speaker gently with a damp cloth.

I would always keep the grilles on just to be safe especially with children or pets around the house it is better to be safe than sorry and a lot of speakers are voiced with the grille on.

I have ProAc D20R   I keep the grills on when not in use.. I think of them as Dust Covers

I had a pair of Apogee full range speakers years ago. The kids begged for a new kitty and I obliged. Came downstairs the next morning to find the kitty sitting on top of the 6’ tall speaker having climbed up to the top putting its claws through the grill cloths and the ribbon tweeter. Long story short- the kitty and the kids stayed but the speakers were sold “ as is” for a $4000 loss!!!so quit gripping about a little cat hair! It could be a LOT worse.

baylinor   I like your thinking, but must digress:  Some music sounds better with the grills on; some doesn't.  My tone controls:  Soft, concave blinds shut, grills on: Least treble.  Hard convex blind side showing, grills on; more treble... brighter with grills off.  Blinds open with tall windows, grills off; major treble.  Windows open, grills off: most natural, and the most neighbors dropping by to chat.