It's worth pointing out that the House passed the "Bipartisan Background Checks Act of 2019" back in February, and that Republican leadership in the Senate refuses to bring it to a vote. I have not heard of any House representatives facing enormous backlash at home over their vote.
As it is, the House is more representative of the population of the US than the Senate, so it's fair to say the biggest barrier to increased gun control right now is the quirky nature of the US political system, not popular sentiment.
