You're right, my narrative was partly misleading. The things that could be most effective like registration would not be supported by most republicans (even though that seems like a fucking no-brainer to me; you register your car, which is an equally deadly weapon). And you're right that mental health screenings wouldn't work, I wasn't talking about that (would be a minefield for lawsuits, like some dude was prescribed an antidepressant 10 years ago and can't buy a firearm... how does a bureaucracy even decide the rules for that? agree about more access to healthcare, which is something a republican congress will never support).

However I think you're wrong about background checks; not for mental health but past convictions, which it seems to be would be simple enough to screen for. This is what the poll I linked was referencing, and republicans do overwhelmingly support it, despite what you think the "gun people" believe. While you're probably right about mass shootings being an intractable problem with regard to firearm regulation, things like registration, safety training, and convictions screening could theoretically help prevent accidental deaths and homicides. It's also worth noting that after Australia largely banned firearms for all but very specific purposes in the 90s, the suicide rates plummeted -- people still attempted just as much, but firearm attempts have way bigger success rates (like 90%) than the average method (around 20-30%).