your quads are not your problem, and leg extensions is a crappy exercise. your problem is in your squat technique. you are lifting your hips before you lift your head. this is causing the weight to shift forward and you are being forced the GM the weight up. this is why you think you have a posterior chain problem. solution? move your head/upperbody with chest out up before your hips move. then move your hips. you'll find the lift to be much easier.
for posterior chain training start doing more goodmornings. no exercise out there to replace it.