I think Democrats are happiest when gun control is not talked about at all. But that’s entirely different from making affirmative statements that they would not seek additional gun control measures, however minor or however downplayed those measures might be. What I think they did try to impress on people is that they weren’t looking to overturn the 2nd amendment or “take everyone’s guns away”, which does seem to be a mistaken meme among a large segment of the right-wing electorate.
Here’s John Kerry in 2004, during his presidential run, saying he wanted to extend the assault weapon ban, which is the key item in your OP:
Even though Bush said he’d sign a bill if it came to him, Kerry is making a concerted effort to position himself to the left of Bush on the issue. He would actively work to extend the ban that got you started on this thread.
So, if you want to claim that Democrats make an effort to counter the ridiculous claims made about them by the right, then sure. They often talk about supporting the 2nd amendment. They like photo ops of them shooting guns. But they wouldn’t have to do that if they weren’t the target of the powerful gun lobby. The same lobby that rated Kerry “F” on gun issues. Not “B-” or “C”, but “F”.
So, if you want to wage a war on misleading memes, you’ll find better hunting on the right on this issue. It’s a target rich environment over there.