This isn't the left's strategy. It's the right's. The right targets these small groups because they know we won't let them be attacked. We will push back. It makes it very easy to paint the left as trying to cater to LGBT folk, but that's nonsense that only sells to those completely out of the loop. Which is unfortunately most of the US electorate. And it's not about it being a strategy. It's about being an ally against bigotry. It takes a really fucked up person to abandon millions of people to increased discrimination because you think it'll help your polling with middle America.
The right capitalized on that because the left was too blinded by ideology to see the harm caused to women.
That's not existing in peace is it, it's an unwelcome assertion of male dominance over women.