Does anyone know why the emergency transporter armbands that was seen more than a few times on TNG...well...at least twice...couldn't have been used to get both Picard and Data back to the Enterprise-E before Shinzon's ship exploded?
They probably could. But I believe the expectation was that nobody would be recovered. There was no particular reason to plan for recovery when Picard expected to die trying to stop Shinzon.
Data didn't jump from ship to ship in the hopes of rescuing Picard, either, but in the hopes of helping Picard fight back a shipful of enemy warriors long enough to blow up the doomsday machine. It was just a convenient coincidence that Data had this miniaturized version of the armband hidden inside his arm at the time, courtesy of an earlier event where transporter recovery was
planned but the armbands would not have been covert enough. It wasn't Data's plan to deliver the transporter aid; it was just something he realized he could do once it turned out that there existed an unexpected way to attain mission goals without requiring Picard to sacrifice his life in the process.
If other E-E crew members had possessed the ability to jump across vacuum, they would have joined the fray, too. If others had been able to transport from E-E to Scimitar
, they would have. But not with the aim of saving Picard's life - instead, the aim would have been to stop the horrors the enemy was planning, and everybody would have accepted that there would be no return. For Picard, there was a return solely because the pair of heroes was successful in eliminating immediate resistance without casualties, and there thus was a crew member to spare for the necessary suicide phaser shot at the thalaron device.
it still doesn't explain why Data and Picard couldn't have hugged and beamed over
The above interpretation does. One person absolutely has to die: the only way to destroy the evil apparatus is to fire a phaser at it, and all sorts of explosives and timer systems are out of the question for technical reason X (chiefly, lack of time). Data just provides Picard with the manpower reserves needed for Picard's safe return.
The interpretation also allows the tiny gadget to be a "realistic" pattern enhancer or beacon rather than an "unrealistic" self-contained transporter. The malfunctions after Picard's departure prevent all beaming, so no reinforcements for the skipper... LaForge's little people then perhaps partially repair the system, but transporting without enhancers inside the Bassen Rift is still lethal. And it's too late to send over reinforcements to Picard anyway.
On the other hand, there simply isn't time (or motivation) for Picard to originally grab an enhancer of the armband type, or to go fetch some explosives from armory, before beaming over. This is also why Picard, rather than Worf, beams over: the likely outcome of the mission in any case is that they all die, the odds of success for the one-man invasion are so ridiculously low overall that Worf wouldn't represent a significant improvement over Picard, and Picard is the one with the ultimate responsibility.