What do you think? Do you see a way to exploit Mobile + Agile to break things?
It does open up any generic character to +20 movement. +10 on 30 gets slightly faster than a wood elf, and +10 on 25 matches the wood elf. +20, on the other hand, far outpaces the current generic character. But, as you point out, there are combinations that can do that already.
My greater interest is that overall among feats, feat powers are unique to each feat. This would create a very rare case in which something other than an ability score increase can be found in multiple feats. It is not game breaking, but it seems to be against the intent in feat design.
A key motive is the potential for narrative distortion produced by Crossbow Expert + Sharpshooter: ignoring cover to deal unreasonable damage at unreasonable ranges (120' to 600'). With Archery and Bless to mitigate the attack penalty. The game becomes one of pavises and kiting.
Great Weapon Master is a gimme for heavy weapon users like Barbarians and Champions. It doesn't automatically produce a bonus attack, and it can't get the +2 to attack that ranged can. It can do a lot of damage, but only in close-quarters. So it is less distorting to our scenarios. And using it to best effect prevents a character from having the best armor class. It does overshadow other melee feats to an extent, but other fighting styles are as strong or stronger... and I believe Great Weapon Master is narrow enough to avoid stepping on toes. It overshadows more because it serves an archetype that appeals, rather than through being the strongest setup.
It's worth thinking about the counter-case of course. Let's say that we do nerf Great Weapon Mastery? What do we gain? We shift the game balance toward casters and ranged. For me, preserving melee as the king of single-target damage is where the overall balance works best.
I don't know much about these feat combinations. Is the basic premise of Crossbow Expert + Sharpshooter that you get one extra attack with a hand crossbow? Nerfing Sharpshooter to once per turn weakens it significantly compared to Great Weapon Master when you have multiple attacks. What if you instead limited it to:
"Before you make an attack as part of an Action or Reaction with a ranged weapon that you are proficient with, you can choose to take a −5 penalty to the attack roll. If the attack hits, you add +10 to the attack’s damage."
This would remove the bonus action attack from Crossbow Expert but retain the Extra Attack abilities.