So yeah, it doesn't use the word, but registration is implicit.
I don't doubt the proponents would wish it so, but I don't see why it's inevitable. Right now, for example, there is no permanent record of background checks (officially, that we know of).
In my state, if one has a CCW that IS the background check. You do your 4473 form at the dealer, and it goes in their file cabinet. Of course, those forms could be collected later and entered into a database (if the law allowed it, which it does not at this time.)
If the NICS check was a go, no go phone call, without a permanent record, then I would have a hard time not supporting that. The problem is that our opponents have completely validated our worst fears about their intentions with regard to banning guns. It's not paranoid if they are really attempting the thing you're paranoid about.