The premise is faulty. Shall not be infringed is the standard.
In theory these things would not inhibit a persons right to keep and bear arms. In practice they would. Universal background checks would by necessity lead to registration. Registration is both intimidating and dangerous. Produce one example of national gun registration that did not lead to confiscation.
Further, universal background checks would almost certainly hinder the transfer of arms between honest people. Imagine the fees and paperwork and the cost. I promise you that the anti-gun crowd is doing just that. Does anyone think that this is anything but an attempt to keep honest people from buying selling and giving firearms between friends and family?