You can do it with some rides, but not all, and thanks to FP, fewer of them. Essentially, with a child in tow that is too small to ride, you have to do the standard procedure for the ride's baby-swap, whatever that happens to be. Sometimes, the baby swap protocol totally bypasses the chicken exit.
It used to be that young children could go through the line at Space Mtn. but bail out with an adult at the chicken exit that's right at the loading area. Now you can't do that; the baby swap is through the arcade ONLY, so no too-short children are allowed to enter the line for Space Mtn.. I don't like this myself, b/c the line area is kind of cool, and I will NOT wait inside an arcade. We no longer swap Space Mtn.; I've just given up riding it.
TOT lets you go through the line, take the child out through the "chicken elevator" and wait in the hallway outside the giftshop, where the ride photo screens are.
At Splash, as there is no chicken exit, little ones are not allowed in the ride line at all; there is a shaded baby playground at the entrance. They give you a babyswap ticket so that after you swap, the second adult can enter the ride through the giftshop exit.