I went into the institute and murdered everyone in sight including Shaun. Then about 400 synth patrollers spawned and I killed them all but I can't leave the institute because a key is required

no. don't rely soley on auto saves or you will be sorry. Save often. If your on console perhaps you have a limted number of save slots and if so cycle through them by saving over the oldest each time.

If you're in console, quick save is very good. When you quick save, it overwrites the previous one.

Find the key then

When I look into my saves file (C:\Users\-NAME-\Documents\My Games\Fallout4\Saves), there's 3 auto saves per character.

No idea how this is helpful.