I very much get what you mean. I like the way Kelly Starrett recently said it (and this is paraphrasing):
"You don't mobilize muscles, you mobilize movements"
I would definitely agree that mobility is movement specific, and the best way to build that mobility is to go through the full range of motion of that movement.
Plan of action, for me, tomorrow:
1) Longer general warm up.
Sometimes I just skip for 2 minutes and I'm done. That has to change.
2) Better use of rolling
I don't need to waste time on certain areas, especially when that time is too short to do anything anyway. I'll spend more time where I need it. IT bands, quads, glutes, bottom of feet.
3) Specific mobilization drills.
I need anterior hip capsule and ankle mobility, bad.
4) Dynamic warm up.
Do it. Don't skip it.
Then, focus on perfect movement during the movements.