The "mobility" part is easier to answer than the "health" part, but if I read between the lines then your answer is TGU for sure. Mobility is essentially the ability to:
A) Put your body / limb into certain positions, and at the same time
B) Having control / strength over those positions
The TGU puts your shoulders (plural ... don't forget the bottom shoulder) into essentially every position / angle of flexion & extension and abduction & adduction imaginable, while of course ensuring you are actually controlling / stabilizing said positions ... in this way it "checks all the boxes" for shoulder mobility better than any other exercise. Other exercises may do a great job of developing control over a CERTAIN POSITION (say a bottoms up overhead carry) but nothing casts as wide of a net as the TGU.
An analogy could be that of a hand grenade versus a sharp shooter: The TGU is your hand grenade ... it solves a lot of problems with just one solution, and is not meant to be "specific." If you still have more things you'd like to address on top of the TGU, select specific exercises to help you "sharp shoot" those concerns.