...he can set it on overload and run back out to that hatch and jump back into space.
Our heroes should remember that in "Datalore", they dumped a Soong android in space, but in "Brothers" he came back to bite them anyway. How could the heroes know that Data was destroyed in the explosion? Indeed, how could the audience?
To rationalize the transporter thing somewhat, what goes down in the puff of smoke is probably the targeting system vital for transporting. Let's say it's all the more vital for transporting through the interference of the Bassen Rift. No lesser system, such as that of the shuttlecraft, can work in the Rift (we know from "Best of Both Worlds" and "Future's End" that shuttle transporters are grossly inferior to starship ones). So a signal enhancer is needed to sort it all out, and the button is a modern, miniature enhancer.
That Data doesn't use the enhancer to beam back with Picard in the end is easily explained: he needs to finish the job, and expects to survive, thanks to being an android (up yours, thalaron dust!). That Data doesn't use the enhancer to beam back with Picard from Shinzon's lab earlier on... Cannot be explained.