Massachusetts already has this.
You can't sell a firearm without a background check. Period. Don't believe me, try to do the whole eFA10 thing without an active FID or LTC. The State rejects it and you can't legally transfer the firearm. You don't have to go to a dealer, but you do have to run a check. The check might be by proxy (valid LTC or FID = pass) but it is the same thing.
Now, so what you say. Well, every six years we in Massachusetts get to enter a legal limbo where our old license has expired but the State hasn't yet granted us our new license. Can't buy or sell in that period. Sure it is supposed to be a maximum of 40 days, but that regularly extends out to months. That is months where you can't buy a firearm or even ammunition. So, you want to buy a new firearm to replace that one that developed a crack in the frame... sorry, you are without the most effective tool for self defense until the State sees fit to allow you to exercise your rights again. Found a good deal on stripped 10/22 receivers? Sorry, out of luck until the State says it is Ok again. State IT workers on strike? Oops, no buying or selling for you.
If I am not mistaken, the background check systems have managed to remain legal because private sales have been exempt. You try to make the case that the government is restricting your right to buy a firearm and they say "go buy private" and you now have a non-interference mechanism to exercise your 2A right. Much like the old "so long as we issue an FID rather than an LTC we haven't infringed on your rights" argument. With a universal background check I think that argument falls. If all sales or transfers are now at the discretion of the government we have literally violated the intent of the second amendment.