I'm right there with you on "happy I don't fall over." I tried squatting while I was pregnant because it's supposed to be good for you, but I kept falling over, so I gave up... lol.
I noticed that I can get "all the way down" in a squat if I tuck my pelvis in. Without tucking, I can get about halfway down to the floor. So I have been squatting that way - really gripping in my glutes to protect my knees - and am finally starting to get deeper into it... not much. But a little... I read somewhere that flexibility is a function of strength rather than relaxation, and since I started thinking of it that way, I feel like I've seen more progress.
Now just imagine a little old lady doing this, lol, and that's my mom.
I do kind of believe it's part of a range of motion, though, which does include a squat like the guy upthread on the pot.
The ones that really showed a good pelvic angle were all of bodybuilders for some reason. But like this