Actually the NFA was challenged in 39. A guy name o' Miller got arrested for having a 17' barreled shotgun w/o the tax stamp. The judge at the trial level gave a one word opinion to the effect that the NFA conflicted with the 2nd amendment & was null & void. SCOTUS heard the case under very peculiar circumstances. Briefly (bad puns always intended) They heard agruments from the government. The defense did not file any briefs, let alone show up to argue the case, as it was on very short notice & the trial defense counsel was doing it pro bono.
SCOTUS remanded the case to the lower court, saying it was beneath their notice that short barreled shotguns were necessary to a well regulated militia & therefore the law didn't conflict. They didn't touch on machine guns or silencers or hear any testimony about how short barreled shotguns do have militia/military necessity. The justice who wrote the opinion was not well liked & his credibility was questionable, so Miller would be very easy to overturn if SCOTUS would ever hear another 2nd amendment challenge to the NFA.
But the NFA is simply a taxing measure. It bans nothing. It merely requires a tax stamp for possession. There are some hoops to get that stamp, such as sheriff approval, fbi check, etc...but anything from 7" barreled shotguns to 37.5mm cannons are legal to own if you go through the hassle & expense. & a registry of those taxed items is in BATF possession.
The GCA prevented any automatic firearms not on the NFA registry already from being added to it. The FOPA prevented any newly manufactured automatic firearms from being added to the registry, in essence creating de facto bans of items not already registered. However The Hughes Amendment to the FOPA was added in the last minutes before a vote on the FOPA & more or less was snuck in that way. It's wording is ambiguous but they interpret it to mean no newly manufactured machine guns. So it's a little unfair to blame Reagan for that one. Blame Hughes. He deserves it.
Coincidentally, in 1939 Miller was arrested for not having paid the $200 tax on that 17" barreled shotgun which would have sold for around $10. Silencers were also subject the $200 tax & that would bring the total price up to about $202, tax included. Helluva revenue raising measure, wasn't it?
& yes, there is a registry of NFA taxed items. However, the BATF is incompotent enough to have scrwed them up rather badly. I've heard that as much as 60% of the records are inaccurate &/or missing. Although the BATF swears they're 100%. So if you do purcahse any NFA weapons make damn sure you hang onto the paperwork, in case the BATF looses their copy & tries to charge you.