Jet Blue and Southwest both do preboarding of families with young children. I believe with JetBlue you may have to mention at the gate that you'd like it. Then, they will announce it when it is time. For both Southwest and JetBlue, IME, certain elites go first, then disabled passengers, then families with young kids, then the rest, as determined by the airline's policies.
As for families not flying on airlines with open seating, that's ridiculous. In some markets, Southwest is the dominant or even exclusive carrier. Plus, Southwest has a provision specifically to allow families to preboard.
And regarding families monopolizing overhead space, this, too, has no basis in fact. My family uses no bin space. We check one or two large Briggs and Riley rolling duffels and carry on a couple of backpacks. We also carry on two car seats and gate check a stroller.
In contrast, when we travel without kids, we always take up bin space, because we carry on our B&R rollaboards.