As I attend more and more births, I find that many women do indeed, at some point, like to be on their back, in a semi reclining position.
I think key to remember is that, in the few hospital births that I have seen, the mom pushed in the same position the entire time she was pushing.
In contrast, our moms, unless their second stage consists of only one or two pushes, usually are very mobile during pushing. They assume a new position on average every half hour. Every hour or so, we encourage them to walk to the bathroom, stay there for three contractions, and then come back and try a different position for pushing.
I think the changing of positions is at least as important as the position itself. The movements involved in changing position encourages the baby to move, the pelvis to open, and minimizes the disadvantages of any position. For example, on a squat stool your bum and labia can swell, especially if you are short, and your legs can fall asleep. On hands and knees, your arms can get weak and your knees can hurt. But if you are up moving around, these things usually don't occur.
Oh, and we don't MAKE anyone assume a new position. but we usually encourage them to get up, stretch their legs, and then say something like "Do you want to try the squat stool, or hands and knees?" If mom says, no, I think I'll go back the the bed on my side" that is fine; we try to get her on her other side, just for a change, but we let her choose. Prenatally we talk about the importance of movement during all aspects of labor, not just the first stage, and so moms are prepared for this.
Just wanted to throw that out there!