It makes perfect sense. I'm just not assuming bad faith on the part of the players. Minimums should not be a part of the game. If someone wants a bonus in the same stat he puts a low stat, that's on him.
The point is if Soldiers need a high Strength to "soldier" then my 8 strength Rogue should not even be able to take that background.
If they don't need a high strength to "soldier" then why is a strength ASI tied to that background?
It makes no logical sense.