That's why I'm interested in what "identity politics" actually means, practically speaking. I get, and share, the aversion to TWAW and lack of protection for women's sex-based rights, but (a) this a relatively recent thing and (b) it's not all that clear that the Tories or anyone else are much better. In terms of Labour's historic support for other aspects of identity politics - the things dating from the 80s that the Mail and Sun take the piss out of it for - what about it is actually so harmful to working class people? Where did the working class get the idea that you can't work towards greater structural economic equality AND socio-cultural equality at the same time? Or that trying to do so is SOOO offensive that it's WORSE than a party that openly opposes any kind of equality and just shits on them without shame?
Identity politics is about dividing up society in terms of different self-identified groups, most often based on categories that are highly socially constructed, somehow rating them hierarchically and seeing the job of politics as being oriented towards serving justice to these kinds of groups. Socially constructed groups means it tends to favour groups based on things like race, gender, and (ever increasing) sexuality categories.
That's not at all of course to say that we don't have things like discrimination based on race. But an interesting change in emphasis can be seen over the years as identity politics approaches have come to dominate - instead of saying, we need a world where being of African or Polish descent is seen as being immaterial other than as an interesting fact about your family, or where your sexual orientation is not worthy of comment and gay people are seen as being just like everyone else, to an approach that almost seems at time to be the opposite of that, reinforcing the distinction of the categories in a way that is almost essentialist.
It's not so much about the exploding of the hierarchy, but the-reordering of it. Except with sex - we have to disappear that because it's not something re-orderable.
As far as whether it is possible to do this "as well as", no, it isn't. For a lot of regular people this is not an ideological statement, it's just what they've observed. Their basic position is stagnant or getting worse while their political representatives seem to spend a lot of time talking about rainbow lanyards and taking down statues. Maybe they would believe they could do both if they saw both happening.
But the identity hierarchy is not really compatible with class based justice or universalist principles. A lot of this came out on the thread about intersectionalism - a hierarchy based on essentialism around race and gender can never really deal with class which will in many cases contradict the supposed hierarchy.
This is why globalists and corporations love identity politics. Adolph Reed once alluded to it as the lapdog of neoliberalism - it allows them to work for "social justice" but protects an economic arrangement that is systematically advantageous for those at the top, of wherever identity group.