This post is so weird, I'm just going to highlight the puzzling parts.
1) The quads are used in the squat until the thighs hit parallel. If I understand this accurately, the quads do not contribute to the squat when the thighs are below parallel. Considering what you know about physics, is it inefficient, in terms of levers, for the quads to extend the knee when the thigh is below parallel?
2) The front squat is fairly important to train for a competitive weight lifter. I am wondering about the extent to which the front squat is a natural movement. It comes close to isolating the quads. Considering a front squat that goes below parallel- isn't is a bit weird to have an exercise which uses a muscle performing an action which it isn't intended to perform? In this case, that action is coming out of the bottom of a full front squat and the muscle is the "quads." Maybe I am underestimating the ability of the posterior chain to compensate at that point in the lift...
This would either be bad (for the knee joint) because it is biomechanically incorrect, or good because it work the quads with a range of motion which cannot be worked in a real squat, or both.
3) It seems like you would prefer to see people press the way you teach the lift, and I have questions about some potential benefits to doing the olympic press and the push press. It does not seem logical for a presser to do a double layback. I do see benefit to having a large hip hyperextension at the initiation of the press. I do not see this a being a disadvantage to the biomechanics of the lift because the initial thrust out of hyperextension is explosive (the levers would be like a normal press for most of the lift). Not only will there be more weight to shrug at the top of the lift, more weight to stabilize with the trunk muscles, and also more work for the traps with the initial explosion, there will be more work provided on the negative; also, might the heavier eccentric contraction provide for a stronger stretch-reflex? And for the same reasons I see benefit with the push press. A benefit which a single lay-back olympic press would provide, which the push press would not, is the work provided for the abs and the superior quads (whatever they are called); that would be a lot more efficient than having to do weighted sit-ups at the end of a workout. Another potential benefit of that press would be that the angle at the initiation of the lift would be more like an incline bench press, without that exercise's limited kinetic chain. It would have potential to do wonders for football players. Also, the olympic press and push press are more power-based in their nature than a press is.
4) As I mentioned, I am not in love with a double lay-back on the press. I did mention some potential benefits to a hyperextension at the initiation of the lift. What would you think about a press where there was excessive lay-forward at the top half of the lift? It would be an exaggerated press- the back would start and end more horizontal, except those two angles would be facing opposite directions. This would avoid the back discomfort of a double layback and also give more room for a shrug at the top (a shrug at the top of a double layback would look like an incline bench press...).
So, there you go.