I've been wondering again and again- is there any actual benefit in setting up bins? Do they reduce the waste that people throw on the floor? Do the doctors feel more comfortable? Anything else?

In general it seems to make my doctors happier, but I am not sure if they reduce waste

I do think they reduce waste - try not placing any in a busier hospital, you'll soon see the floor covered with soda cans, banana peels and other stuff.
(At least, that's my experience.)

Well , the bigger the room and more stuff in it the more happier the doctors will be, bins are REALLY cheap and I think they reduce trash, I think I am in the 5th or 7th stage and I put A LOT OF BINS (2-3 bins in each room), the hospital is jam packed, my doctors are happy plus there is little trash for the amount of people present.

I think that improves the rate when the VIP checks the room.

It's a great curiosity question. My answer is they do nothing; it's just a decor item. Trash is caused by an annoyed patient or at random rarely. That's after 20 years of playing the game. My advice is if it makes you happy to place one, do it. If you're indifferent, save your money and use it to make a return. It's a business game with an element of drafting.