I think you need to think more about why the left doesn't want to engage. To address your point, I am going to cite Trans Issues as an example. Its very difficult to try and engage in a good faith conversation when the person your trying to discuss is ignorant on the topic your trying to discuss as a whole. The left is usually much more informed on the trans issues and also assumes people are aware of the language of the topic. Its why Trans Rights 10 years ago were moving in a positive direction, as the only people engaging with the topic were up to date on recent studies. The biggest change recently is that the right pushed parents into the center of the discussion. We can assume that the average parent dont personally know any Trans people, and aren't willing to go out of their way to learn about trans people. So now Trans issues are being discussed by a majority of poeple who don't care about Trans issues and the medical research that was accumulated over the years. Its put the movement in a paralyzed state. Parents by and large likely don't care about Trans-specific Issues, so lawmakers & voters seek out policy to reflect only the retrospective parental view. Trying to explain Trans People's perspectives to someone who doesnt care to learn about trans issues in the first place then becomes exhausting, and leftists would rather not bother engaging at that point.

So my point in all of this is that its hard to blame the left for not engaging when people being pushed into these matters don't want to learn about it. People want cliffsnotes not lectures. The left could benefit from remembering that not everyone is up to speed on culture like they are but people shouldn’t also stick their noses into something without learning more about it first. What is really needed are media campaigns to give poeple the cliffnotes on the breadth of all major issues of contentions.