I have a quest that requires me to reforge an amulet, and one of the pieces is in this dungeon but it has "requires key" so I can't enter. Any would be appreciated.

Try College of Winterhold?

saarthal is a location of the first quest you do for the College of Winterhold.

Thanks. Is that the only way, though? I'm playing a warrior so the college isn't of any interest to me.

The quest starts even if you don't enter the college and can progress till Sarthaal. There are no obvious hints that going to the college is needed (the excavation site is empty) nor you can ask around to learn that a mage is conducting researches at the ruins. Quest is basically borked for non mages characters

If you're on PC, open the console, and type unlock in front of the door. The original quest (I think it's called "forbidden legend") will progress just as fine.

The first real mage college quest goes there. If you have no interest in the mage's college you can just stop with that one - or get expelled by punching Toldfir in the face (and running away very fast).

Afraid so....one of the unexplainable directions the devs took with this game.

Yup, the only way to finish that quest is to join the college