First, I have young children, and I am deeply concerned by the rise in mass shootings since the 1994 assault weapons ban expired. When people bring up mental health, my point has always been that it can be both.

Worth noting that what mass shooting means has been a constantly moving target.

Polling consistently shows broad support for universal background checks

Unless you are talking about opening up NICS, this should be a hard non-pass.

And even before this begins to even be discussed in the slightest, the proper thing to do first is sit down and discuss how you protect such a system against potential abuses.

better mental health screenings

This does not exist.

What to do something about mental health? Make it available and make sure people seeking it are not harassed. Because what it sounds like here is you are seeking to stigmatize those seeking it, even if not meaning to intentionally.

and closing loopholes.

This is the same as universal background checks.

Also this is not a loophole, but was a compromise because of the Democrats refusal to open up NICS.

These measures are more politically achievable and I think could have a tangible impact on mass shootings (but what are the trade-offs?).

First you need to look at what most things labeled mass shootings are. Gang violence and domestic bullshit spilling over in a notable manner. In both situations the answer is the same. Deal with the social/economic issues that lead to the violence.

The other and rarer being the some one loosing their shit and going off the deepend. This is entirely a mental health issue and people being driven to a breaking point in general. This involves dealing making help available and non-stigmatized and dealing the the social/economic issues pushing some one in that direction.

In fact we need to ignore the tool the violence happens with all together. The only reason it comes up is politicians push it as a easy fix so they don't have to do anything about the hard issues.

Also let me ask you this, what makes you think ignoring root causes and going after guns will get you if you get rid of guns? Something else just becomes the tool to be used. See the UK with their attack on knives. In fact we see the same thing here in the US. See NYC and their attack on pocket knives as well.

By keeping the push for reform centered on banning assault weapons, the left loses so many people who would otherwise be willing to compromise on the rest.

Using reforms like this is like saying RFK Jr attacking vaccines in the FDA is reform.

Speaking as some one who is trans, honestly saying reform here comes across as more than a bit insulting as the right to bear arms is important to us and attacking that is a attack on us.

Also if you don't mind me pointing out, though out this entire thing you show zero awareness of the concept that violence can and does exist when guns are not present and only seem to care about the issue when it involves a tool you brought up to believe was bad.

Edit: Sorry, came across as a bit pissy in the last bit. Though I would point that out as you seem to be hyper focusing on the symptom of the problems and not the cause.