Go after the ankles, and go after them hard.

The fact that you’re able to squat more upright in the Oly shoes points strongly in that direction. Since they wedge your heel up, they take some of the mobility demand off of the ankle. More hip mobility would help you further compensate for that lack, yes, but that probably isn’t the best way to solve your problem.

It can be frustrating, and they can take a lot of consistent effort to improve, but things will pick up. Don’t stop until you can perform that pistol switch without your Oly shoes.