Plumbing – Why would the bathtub spout stop leaking after fixing a water pressure issue

Diverting the leak up to the shower head using the diverter valve should not be a problem, but your assumption that fixing it will be "pretty easy" may not prove to be true. You are correct in that hot and cold supply to the tub/shower valve needs to be shut off. Sometimes apartment shower valves have built in shut-off valves called integral stops.

Sometimes apartments are equipped with individual remote shut-offs. Sometimes the whole building will need to be shut down. Removing the valves may require special tools like deep valve sockets.

The repair may be simple washer replacement or may require more complicated cartridge replacement.

I do recommend that you try it but make sure your landlord is cool with it in case something breaks, and have back up shower facilities available in case it ends up being out of service for a while.

If they aren't hexed holes then they aren't replaceable, you might have to put a hex bit in to confirm removability. It sounds like you removed too much. If the seats were scratched, pitted or therefore defective, then you should've put new good ones in instead, if possible or returned the taps. But, you shouldn't touch the seats unless or until they leak & if you do it should just be a light few turn honing.

If the seats started out like most seats, being proud or raised from the shoulder, but are now flat or even with the shoulder. Then, you took way too much off. Only the seat is your seal, not the entire shoulder. You'll either need to get new seats, if they remove. Or, have to go to thicker washers. Or, get new taps with replaceable seats & not touch the included seats.