Clover is a very different plant from bermuda grass, although the clover can get invasive as well.
At The Farm, I just let the clover grow; I don't mow it. It can get 18 - 24 inches tall during cooler weather. It develops a nice duff underneath it. I'm not sure how it would work for a lawn substitute. For one, if you didn't mow it, the height would probably classify it as a weed for purposes of Phoenix zoning ordinances, and if a neighbor complained you'd have to battle the City over whether the green patch were legal or not. I assume that, if you mowed it, it would develop a lower, tighter habit that might look something like a lawn, although probably less even-looking than a conventional lawn. The clover suffers a bit toward the end of summer, particularly if it doesn't get enough water or is too exposed to the sun. Also, as I remember from being a kid in SoCal where clover grew in the grass in parks, clover attracts bees. If your kids are playing in or running on clover, they're going to get stung eventually.
Like everyone, I have constant battles with bermuda grass. The clover keeps it more or less in check, but doesn't outcompete it. Because I don't mow the clover, what happens is that the bermuda invades in a very tall, lanky form and kind of co-exists with the clover until I rip it out. I suspect that you'd get a different effect if you mowed, and mowing might give the grass an advantage.
There are a few weeds that can come through in the clover. The first I deal with is cheeseweed. This one you just have to pick after irrigation and before it goes to seed--after a year or so of conscientious maintenance it will essentially be eradicated. The second is bermuda, especially around the perimeters. In its lanky state, it's relatively easy to pull from unmowed clover, although it takes a sharp eye to get it all. The third is nut sedge. This one is the hardest to deal with and will outcompete clover in sunny locations. I had real problems with this, and covered the worst nut sedge patches with cardboard & mulch to kill them. Then a few years ago the nut sedge just stopped being a problem. I suspect that, as the trees grew and the ground got shadier, this gave the clover the advantage. Johnson Grass will also outcompete clover, but I believe Johnson Grass is relatively easy to eradicate, if you're willing to get out and dig in the hot August sun. Gratefully, Johnson Grass isn't that common.
Short answer, no. I don't think clover would make a good lawn substitute in a conventional lawn setting.
There's a strange weed that's growing in my bermuda lawn at home. It's a dicot, and creeps very close to the ground and has leaves about 1/2 inch long, shaped a bit like chinese elm leaves. It's not at all like clover or medic--it's tougher than that. It has a very small yellow flowers, and does better in shadier portions of the lawn. It's kind of nice, really, and holds its own against the bermuda.