That's always been my interpretation, that the "personal transporter" was just some sort of remote control or beacon for the ship's machinery.
It should be noted that the concept of a personal transporter was introduced quite a few years earlier in Voyager
's "Non Sequitur," though there it was a rather larger device than the one in NEM. There as well, I choose to interpret it as a device that connects to an existing transporter system and takes control of it, rather than one that actually somehow teleports itself.
Hmm... I suppose a self-beaming transporter could work if it had two redundant transporter mechanisms -- transporter A would beam transporter B to the destination, then transporter B would beam transporter A there. But it's hard to believe that even one transporter could be miniaturized to the size of that tiny device in Data's arm, let alone two.