They have intentionally broken the boarding system in order to get people to pay for upgrades just to get that baggage space. During covid they boarded from the back of the plane to the front, and it was gloriously efficient.
A lot of commercial planes are tail heavy, loading a bunch of people in the back first is going to tip it. Doesn't even need to tip all the way to cause damage. It's a scary thing to see in person. So yes it does matter what order people are seated in.
I think every airline who gives you a free carry-on will check it at the gate for you for free. In fact, they'll often beg you to do so. But at that point you're losing the benefits of carry-on baggage like access to your stuff during layovers, getting to skip baggage claim, and no chance for them to lose your stuff.