Does the subwoofer have to be on the floor?

It certainly will with drowers!
All I can suggest is to remove drowers and put sub in that space standing on floor as far as my imagination allows to picture your library.

I can remove drawers, but I will still have a shelf underneath that I cannot remove. For that matter, I think there is some hollow space between the shelf and floor. Would simply removing the drawers be good enough? I do not understand what the big deal is anyway. Regular speakers can have stands, why can't the bottom shelf, which is very solid, act like a speaker stand?

...the hollow space is not a big deal and you can use it, but since sub has a large flexibility of placement i.e. can be realy away from the mains or somewhere in the middle of room or under the journal table or even under your desk.