and therefore there is no need to go back.

If you are paying by cash, most hotels require a deposit beyond the room cost for incidentals, and then you do have to go back to settle.

I suppose you could always tell the hotel that your plans have changed and must leave early because of family emergency, etc.

Don't try to pull that more than once at any hotel in a given year, however or they will suspect something fishy.

No, you settle the bill when you check out.  You paid when you checked in either ahead of time, with cash or with a credit card. Unless you need a receipt there is no reason why you can't leave when you are done with the room. I have been doing it for years...