Honestly, I think I stated this backwards the first time. (It’s been fixed for anyone else looking at this post)

If your hips stay solid with bracing on, it IS NOT a mobility issue.  You can achieve proper depth with a stable midline and have all the ROM required to do so. 

If your hips tuck under with bracing, it IS a mobility issue.  When you achieve proper depth you go into posterior pelvic tilt (butt wink) and this is what is shutting your abs down, in my opinion. 

Try doing the same movements as a body weight squat and see if you get the same results.

Overall, you first need to be organized and extremely stable under load.  The WHOLE time too.  If you are losing stability in the bottom, make sure you are really screwing your feet into the ground and maintaining that initial tension.  Secondly, you need to have the ROM to perform deep squats.  You have already pointed out what needs the work; anterior hip, ankles, and hips.  For now, I would squat higher, to a box or free, until you can fix the butt wink.  Then add a little more depth when you are sure you can maintain neutral.