This is a fundamental problem within the left-wing, and can't adequately be addressed by either the right or the centre. Those already on the right and centre will not be enough to persuade normie left-wingers to understand the virulence of the woke left, and how their beliefs and actions are antithetical to established social democratic philosophy. Normal lefties don't want to abandon their left-wing principles solely to stick it to woke lefties, and they shouldn't be expected to; there's more to politics than that. And this only serves to benefit the woke left in the long-run sadly.
The reason woke leftists have attained so much societal clout as of late, outside of mass corporate astroturf, is that the alternatives have encouraged abandoning the left-wing entirely for their own gain. You don't have to be a right-winger or a centrist to abhor the woke left; and by declaring that you're "leaving the left", you allow the woke left further validation that the rest of the left supports their behaviour. If anything, the behaviour of the woke left is eerily reminiscent of the puritanical Reaganites of the 1980's; the kinds of people who would decry heavy metal as satanic worship, and an affront to civilised society.
The left is in desperate need to remind everyone else that we do not stomach cancel culture or intersectional theory. That's the only measure we really have left in getting more normal people to stand against the woke left; rather than the perpetual right-wing apologia or outright racial discrimination that only serves to vindicate the woke narrative against everyone who speaks out against it. We need less rants about "muh jewz" and "muh blackz", and actual focus on the avaricious, hyper-capitalist corporations and affluent individuals that benefit materially from creating disarray between the citizenry and profit off human misery, as well as demonstrating the unwarranted self-importance and puritanism of the woke left.
We need a strong left that values socially liberal and legitimate progressivism to truly address the aberration that is the woke left.