After reading all of this I'm afraid I need a new saddle... Is there anyway to tell from looking at a picture if it the saddle that is putting you in a chair seat?
Probably. Sit in the deepest part of the saddle. Put your feet in the stirrups, and move your legs until the stirrups hang straight down. That is the position gravity tries to put you in. Lunge a horse with a saddle, and see where the stirrups go at the trot and canter - pretty much straight down, thanks to gravity.
If it is a chair seat, so be it. I suspect most saddles - unless you are like my daughter, riding in an adult saddle because that is what we have even though she is 13 - will not put you in a horrible chair seat. Not unless the saddle is WAY to big for you...like my 5' 0" daughter riding in a 16" western saddle.
I have tended to ride in a chair seat because it felt more secure to me. And I think I felt that way because in a sense it WAS more secure for me. I have very tight hips, and taking up riding at 50 was a real challenge. If I tried to bring my heel in alignment with my hip, it required a LOT of tension in my leg - unconscious muscle tension.
That tension made riding with my heels under me like riding with pine boards for legs, and that threw me off more than a chair seat did. A chair seat allowed my legs to relax, which was more important than perfect heel position.
Just this last month, I've found it possible when riding my <900 lb Arabians to bring my heels under my hips without tension in my legs. And like the riding instructors will tell you, that improves my ability to sink into the saddle.
But 3 months ago, it simply wasn't happening. Not with my body and my tight hips.
I thought this comment in the OP was interesting:
"The big issue here is she is so so stable when riding with this position...She does all this bareback too!" That makes me think that the individual rider - width of hip, flexibility, length of thigh, etc - might NEED a chair seat to fit her horse. Even now, when I get on a wider horse, I find my legs moving forward.
I think heel-hip alignment is kind of like toes forward - a good idea if your body can do it, but not as important as riding with a relaxed leg. If you ride with a relaxed leg and hip (and don't lean your torso), your rump will go to the deepest part of the saddle (thank you, Mr Gravity) and your legs will drift towards a vertical stirrup strap - but only as far as your body type, flexibility, saddle and horse shape allow.