That's a good specific example, universal background checks already exist, unlicensed dealers don't have to do it because they're unlicensed. The bill the House passed that people are pointing to as a solution to everything requires that unlicensed dealers give a gun to a licensed dealer who then does a background check before the unlicensed dealer can turn it over. This is a strange response to the recent two shootings because not only wouldn't it apply, they bought them at gun stores, but both would have passed background checks anyway (which is how they got the guns obviously) as they didn't have any problems in their background. The other major proposal is to add mental illness to the background checks, which would be a massive expansion into private medical records to cut people off from rights they'd have without a documented mental illness. Courts have come down hard on these things already on issues outside of guns. If you have mental illness to where you have your capacity taken from you, you can't buy a gun already. As an example, Laura Loomer (perennial GOP candidate of handcuffing herself to Twitter HQ fame) recently tried to claim she was being denied a gun license due to her conservative political views but the document she posted revealed that she had actually been denied because she had a court order against her finding her not competent due to mental illness.
I am not saying that these measures may not be good to pursue, nor am I saying that the recent shooters shouldn't have possibly received mental health care or other interventions, just pointing out (as I am wont to do to everyone's annoyance) that the narratives don't make any sense.
I will admit that I do think there's some kind of comforting response people are doing where they pretend you just pass this magic super supported legislation (the details aren't important) that will file off these extreme outlier events that exist outside of the "normal" American culture of violence that most people treat as background noise despite the significantly higher risk profiles compared to the more dramatic events without having to dwell in the complexities of needing to address violence in general and hundreds of millions of existing guns that don't have acceptable easy answers yet people want solved to their satisfaction perfectly immediately. Assume for the sake of argument that this House bill would completely eliminate mass shootings of the Buffalo and Texas varieties, it wouldn't make a dent in gun deaths or violence in general yet the way the issue is presented and treated is so dramatically far from that reality. I find it interesting from a political science perspective, I'm not begrudging anyone just emoting in response to tragedies and crimes even if I find some of it on Twitter to be quite amusing.
For example, I was just looking at some tweets from "prominent" tweeters screaming about how Chuck Schumer is a fascist murderer of children because he recognizes that he doesn't even have 50 votes in the Senate to repeal the 2nd Amendment and order all guns to destroy themselves. Most of the people seemed to be angriest about the fact that he's not bringing such a bill to the floor of the Senate so it can fail to pass and potentially harm his own Senators. Plus that he suggested such an action wouldn't be worth anyone's time and that they should try to build support for what they could possibly get passed and hope the midterms might give them more seats. Schumer's crime seems to be that he acknowledged reality rather than "fight" (lots of usage of this term) as if this would somehow demonstrably alter reality itself towards what the tweeters want. A romantic symbolism seems so essential to where it overwhelms everything concrete.
Politics is a strange beast. Quite frankly anyone interested in it is some kind of disgusting sex criminal.