"Infringed" can certainly be read as any process that gets in the way of gun purchasing.
[0] Except in cases where you have a CCW (which you go through a background check to get already), a purchase permit (many states require a check from the sherrif or similar office before buying a handgun) or some other similar stand in that does something significantly similar to the NICS checks.
It’s not a “maximalist” interpretation. With nearly any other federal right, we’d agree that maintaining a registry of people exercising that right would impermissibly create a chilling effect. You can be a member of a minority religion, but only if you place yourself on a registry! You can exercise freedom of speech or assembly, but only if you put yourself on a registry. It’s not maximalist, it’s giving an enumerated constitutional right a full and fair reading.[1]
I do agree that the existing precedent in the area has been impoverished. The Supreme Court has important work to do correcting that.
[1] The framers were gun nuts and would have been troubled by the decline of gun ownership. The idea that you have a civic duty to own firearms would be closer to the original intent than the idea that the government can limit gun ownership to formal militias.
> Before a standing army can rule, the people must be disarmed, as they are in almost every country in Europe. The supreme power in America cannot enforce unjust laws by the sword; because the whole body of the people are armed, and constitute a force superior to any band of regular troops. – Noah Webster, An Examination of the Leading Principles of the Federal Constitution, October 10, 1787
The US doesn’t keep registries of religious buildings either. There’s obviously titles and whatnot for buildings. But it would likely be deemed a constitutional violation to require registration of every place that would be used to hold religious services. Some in Germany have proposed a “mosque registry”: https://www.dw.com/en/german-politician-wants-islam-law-and-....
A much more accurate analogy might be a registry for religious paraphernalia (crosses, rosaries, etc), including information about the purchaser. Such items are tied to an individual, and thus serve as a very effective proxy.