Universal background checks are another way of saying no more private sales -- at least in the sense that they are private.
The real problem: To enforce universal background checks, you would create a system of de facto registration. The problem with that? Once registration is in place, confiscation becomes doable.
Imagine we are 10 years down the road on universal background checks. Feinstein 2023 passes, with a ban on "assault weapons," a six-month mandatory buy-back and no grandfather clause thereafter. That law is workable because they have 10 years of records to use to send out notices/summonses/agents to those who own the weapons. No need to search every house on the street. You get a certified letter saying turn in AR-15 No. 1234567 within 30 days.
As it stands now, confiscation isn't really considered, in part because it's logistically impossible, because there really is no telling who owns what. (Even if they really are keeping and logging all the 4473s, there's millions of private sales untracked.) But after a decade or so of de facto universal registration it looks much easier.