school me on batteries and amp hours

I understand why amp hours are important to deep cycle situations where the battery is not connected to a continuous charging system (solar applications, ‘total loss’ vehicles with no alternators). But why are amp hours important to starting/cranking batteries? Sure you need enough amps to start the engine, but after that the vehicle’s charging system kicks in, so who cares how many amp hours it would run without charging? I don’t see how that matters except 2 rare conditions: if you accidentally leave the key on and also if the engine is having trouble turning over and you have to crank longer than normal. Other than that, I am failing to understand why starting batteries make a point about amp hours or anything other than cranking amps. What am I missing? thanks