As far as you not feeling quads much in your heavy work, thats because your hips shoot back (and I think) allow your weaker quads to extend so that you can get tension in your hamstrings and finnish the lift with a goodmorning.
Now when I said quads that was a little missleading of me. Considering glutes also play a large role in getting you out of the hole while still staying somewhat upright. Which is why I mentioned front squats. All though they hit quads real hard there is no way anyone is going to get out of the hole without using allot of glutes as well.
But all this stuff is just theory from me. looking at your form I would say front squats would be the best thing you could do to raise your squat and fix your form some what.
I know theres allot of westside shit going around saying quads dont do much in squats etc. But for raw lifters squating deep with a medium stance there is allot of quads involved. You probably know this but Im going to post it anyway so some westside newbie doesn't come in quoting louie simmons on me.