I think that on top of weak quads, glutes could have something to do with it as well. A lot of people have weak glutes, plus movement patterns that don’t fully involve them. When your legs are bent like they are at the bottom of a squat the hamstrings are in a lengthened position which means that they can’t produce much force. By shifting into a good morning you are putting yourself in a position where the hamstrings can take over. If the quads were strong enough to extend the knees and the glutes were strong enough to extend the hips then that wouldn’t be happening.
Other than just looking at weak points, the other thing to look at is technique. If you have this problem to some degree even with weights well below your 1rm/far from failure then half the problem could be that you have bad technique.
If I was you I would stop pushing past technical failure because once you have enough weight on the bar you won’t be able to finish a squat as a good morning. Some quad and glute work would probably pay off too.