Capitalism was about economic ownership by capitalists—the class of people at the top of the economic hierarchy, who are wealthy enough to start their own businesses or buy shares from others and earn money without working

I think that’s only true because in early capitalism, only the existing wealthy were able to participate in the system. That’s still the case to a large degree, but it’s not an intrinsic feature of capitalism. It was just an incidental aspect of capitalism in the world as it existed.

True capitalists believe that the more competition, the better. Giving ordinary workers a higher stake in their company promotes a much broader level of competition. I think it’s totally fair to describe worker co-ops in terms of capitalism, and as you said it would really help the branding.