if he overcommits
That depends on their faults. Meaning, you have no say in it. They just dance right outside the turret range and can still dmg you from it. Cait can dmg you even when you're under turret. I play Darius and the turret has no impact at all. I just Q whenever they get a minion. If they do the "Smart decision" of giving up cs, they end up with even less cs than their jgler.
IF you are low health and still stay under turret to get dove,
Because there's nothing to do. There's no counterplay. Because of how snowbally the game is, they will just get more xp/gold than you every second you're under the tower. If you retreat, you have a gap that can never be closed, and you'll be useless the entire game. Based on how weak turrets are, if you've been pushed, the turret will likely die too.
I know you are supposed to protect the turret, but it falling down isn't the end of the world
It opens up first blood, as you mentioned. It gives them roaming potential, they can basically dive you whenever they want now, as the wave will always be near the first fallen turret.
If you think about it "strategically," no matter what you do, you will be useless the rest of the game