From what I see, it's not at all the same as the English (dressage) approach. Their horses are taught to keep off the bit ("give me your face" is one way I've heard it). As for coming under, that happens because the Western horse is asked for more effort (speed) right from the start. The horse finds what is the best way to carry himself. (While keeping off the bit.)
This actually isn't true. When we train a western horse you do train them to accept contact and move properly. When I'm training my guys I start just the same as you would english. I use all of my aids, leg, seat, balance, and hands. I can pick up my western horse (in a western bit) any time I want to. The reason it looks like there is no contact on the finished western horses is because they have learned how to respond to very slight changes in hand possition and varying degrees of contact.
If I'm asking my WP or reining horses to frame up and give me a good head set I lift my hand and put it back a little to get the contact I need so my horse doesn't speed up. I also put my legs on and drive with my seat at the same time as picking up my hands.
A western horse doesn't learn where to hold their head just because we give them a long rein. They learn where and how to hold their head and how to engage their body by practice at home using a lot of the same technics of english riders.
And "give me your face" is an expression used not because they don't want the horse on the bit, but because they want the horse to soften to the bit. They want them to work with the bit, hold it, carry it, and move using everything that the horse is.
Most western horses are not asked for more "speed" from the get go. Heck, in most western diciplines we want them to learn "slow". Even when training a horse for gaming we ask for correctness if done properly. A horse can't turn a tight barrel if they aren't moving correctly. You need to put on the buttons before asking for things at speed. Just like you need to put in the fundamentals before asking a horse to jump high.
A horse can run quite fast withought being collected. In fact it's usually when you aren't asking them to move from behind and use their haunches that you will get a horse that runs faster. On the other hand, a horse can't move as well really slow if they aren't collected. They will run through their transitions if not collected and balanced and they have a much harder time maintaining a slower lope if they aren't collected and balanced.
Hope that helps. Sorry if that was a bit of a rant.