One way an independent publisher could use FLoC is for "lookalike" targeting. For example, if your best customers often have IDs D4W1 or JQ82, you could choose to target ads at those IDs.
Some entity might choose to run a "store" like you're describing, but there's no reason there would only need to be one.
It looks to me like Chrome is proposing a low-level feature that anyone can build more sophisticated products on top of.
I think I have an idea. So by visiting site A, A gets your cohort and now there is somewhere a "central store" or register. Using that A can now do ad bids for that entire cohort to reach others.
But where is that central store? Is Google running that?
Might also work fundamentally different.