More quad involvement happens when you have the most flexion at the knee. If you don't have the mobility in your ankles to allow your knees to fully bend then you either:
- Come up on your toes or
- Fall back on your ass
Obviously, you don't want the latter so oly shoes essentially allow you to come up on your toes while maintaining a stable base.
There is less strain on the knee by raising your heel but can you see why you don't need plates/shoes if your ankle mobility is great?
As for shoes over plates:
The shoe has a totally flat surface giving you more contact with the floor (more stable). On plates, you get reduced contact points
Shoes have no give which means there is no power loss. I'm assuming you are wearing some rubber soled shoe?
Finally, all your focus is on the squat and not trying to find your footing or worrying about missteps.
None of this is a big deal if you are using moderate weight and just doing it until your mobility improves but you may consider them if you're always using plates.