"Common-sense" is a highly opinionated phrase. I do sympathize that it's hard to find a good debate due to the misinformation or poor arguments on both sides. Without good debate it is tough to get a good view of the issue. Even then, there can be points that come down to philosophy or law around how the action can/should be implemented. And of course, there's very little nuance or compromise being investigated by either side.

That said, which LGBT rights do you think they oppose? Do they oppose all rights or just "common sense regulations" (see where I'm going?)? What one person sees as a right, someone else might see as a privilege. Or what they see as reasonable regulation another might see as infringement. The sides have to convey their reasoning for why something is a right, or why they see it the way they do. It's supposed to be an effort to understand each other to make things work for both sides through compromise.

There are a lot of vehement and downright ridiculous comments out there. Perhaps you see more of them on the LGBTQ issues. Perhaps gun owners notice them on gun issues. So perspective can matter. You have people like Beto O'Rourke saying they will confiscate certain arms. There the DNC committee member that said she thinks nobody should have a gun (Bonnie Schaefer). Many have declined to answer whether they believe if owning a gun is a right, including Hillary Clinton. If they can't answer the question, then people will think it's a bad sign. There are a lot from Feinstein that I'll put at the end. And this doesn't even cover all the misinformation (what was used in a specific crime, things that are already illegal), frivolous laws (a folding stock isn't making a difference), and wrong terminology (assault rifle, clip/mag, etc). It's hard to write intelligent legislation if the terms aren't used correctly or the subject isn't well understood. I assume you've seen the same on cis people writing LGBTQ laws.

Many gun organizations have supported past restrictions. Maybe we could implement "universal" background checks by creating a free to use system that can be used for other types of checks as well and doesn't keep a record of what the check was for (maybe using a zero trust pattern or some one way hashing which industry/watch group oversight)? This would address the concerns the concerns of building registries or abuse of power while making a seller civilly liable if the person they sold to was prohibited. Why doesn't anyone in power suggest such a thing as a compromise?

There are tons of options out there. "Common sense" has just become a marketing term by politicians on both sides to appeal to people without actually proving out their issue.

I'll leave you with some quotes from Feinstein:

“Banning guns addresses a fundamental right of all Americans to feel safe.” – Associated Press, 18 November, 1993.

“If I could have gotten 51 votes in the Senate of the United States for an outright ban, picking up every one of them; “Mr. and Mrs. America, turn ‘em all in,” I would have done it.” – 60 Minutes on CBS, 5 February, 1995.

"All vets are mentally ill in some way and government should prevent them from owning firearms."