what does this "culture" actually offer the rest of the working class?

Anarchism. Although I understand that that term means different things to us so I'm going to use the meaning you gave it a few comments back:

Anarchism consists of a critique of hierarchy

And this cultural anarchism is taking that critique and applying it to culture. To everyday situations. To the way children are being raised and workers are being hired. To song, writing and all the other arts. What it offers to people is anarchism. A way to live your life without archy. Or as AFAQ calls it: "social revolution" https://anarchistfaq.org/afaq/sectionJ.html#secj7

This is what I mean when I say revolution: A complete change to the entire social structure. The biggest driving force in any society is culture. While economic forces to play a part they can only exist as long as they are reinforced by culture. The value of money exists in culture. The concept of property exists in culture. An anarchist culture is about looking at these concepts in a way that consciously opposes archy.

Also participating in capitalism does not yet disqualify you from acting anarchically. It's not a all or nothing scenario. You do what you can, where you can. Obviously you should be on the lookout for better alternatives and constantly keep in mind what it is your participating in every time you shop, but as long as your thinking about it, considering your actions in an anarchic framework, you are acting anarchically.

And the more people keep doing this the more they start considering alternatives, at which point anarchic spaces become a vital component to in the process to collectivise the economy. You connect people with skills who don't like having to shop for food and some of them might start their own farm, and because they already have connections to other people in that space they start being able to benefit from that venture as well.

The social/cultural isn't separate from the economic which isn't disconnected from the political. Society is a collection of all and in order to effectively dismantle one we need the help of others. And culture is the easiest by far because all you need is for people to listen and consider the things you say. Culture is nothing more than the ideas we hold and ideas are a hell of a lot more easier to change than political or economic realities.

But that's just the framework that I use to think about anarchism and society at large. You probably have your own.