They can't.

Center/moderate left ideas work, but the policy proposals they have inherited and their base have become emotionally invested in don't work. Some bad, but highly valued policies rely on the far left's assignment of authority based on identity to circumvent debate.

If someone says "White people can't understand black issues, therefore this black author gets to tell us all about what what people think, feel, believe, their connection to white supremacy and how racist individuals are across the country, and reinterpret their history" you recognize that as woke blather.

If someone says "Republicans just think climate change is a hoax, listen to the climate scientists!" you don't immediately recognize the same issue because of the straw man about climate change being a hoax, but when you watch a debate on CSPAN or something where a republican starts talking about cost vs benefit, suddenly you see why some (not all, but some) wildly expensive and unproductive climate proposals require the same sort of deference to an authority. Without "This guy is an expert and he likes it so we shouldn't have to talk about what it costs!" some key policies just don't survive, and the center/moderate left can't give those up because it would be a betrayal of their base who are emotionally invested in those policies.

If you allow authority by identity to circumvent debate, you de facto endorse the woke crap because that's how it got it's foot in the door in the first place.