As they should. That weight sensor is not intended to make it safe to put a car seat in the front of a car.

So what's the difference between a front seat without the airbag on and back seat?

My guess is that it's really just automakers CYA so they don't have to treat it as a safety critical system. There's really no reason to have that sensor then.

It is part of a safety system, but the other way around. One of the purposes is to make an annoying dinging sound when the passenger does not put their seat belt on.

Eh, if one doesn't want to wear a seat belt, then don't. Pretty annoying feature if you're using it as a farm truck.