Not having the chains would actually slow the game down. Remember that for every time the chains are brought in, there are probably 10 times where the first down is awarded (or not) based on the position of the sticks, without bringing them in from the sideline. If you're a yard ahead of the first down, they don't bother to measure. They just signal the first down and move the sticks. If you're a yard short, they don't need to measure. A measurement is often requested, but the officials may refuse.
With a chainless method, they'd have to measure every time. Think about it: no indication anywhere on the field where the first down marker is. They would pretty much need to measure after every play.
It would also be a huge disadvantage to a player who is fighting to get a first down. "Do I fight for extra yards, risking a fumble, or can I just step out of bounds or go down knowing that we've got the first down?" And a lot of pass routes require that a receiver go just past the first down marker and make his cut. With no first down marker, you wouldn't be able to do this anymore.
I like that you're thinking outside the box (no pun intended...okay, maybe a little bit intended), but until they can come up with an alternative that addresses issues like that, I think we're stuck with what we've got.