f they can determine your browser was associated with some set of unique IDs, then they can track you

The only scenario in which this could happen would leave both strategies equally vulnerable.

No, the idea is that you can’t be traced via fingerprinting.

Both strategies accomplish that.

You and 1000 friends go to a party all dressed in the same Mr Blobby costume. When one of you gets absolutely shitfaced at the open bar and vomits in the middle of the dance floor, they get kicked out and banned from next week’s rager. Next week rolls around, and 1001 Mr Blobbys rock up on on the dance floor, because management has no idea which Mr Blobby cost them their deposit last week.

You and 1000 friends all go to a party dressed as a unique DeviantArt Sonic OC. One of you fails to hold their liquor. They get kicked out. You all attend the party next week all wearing a completely different costume of a completely different DeviantArt Sonic OC, since the number of them is functionally infinite. Management can’t kick the vomiteer out because as far as they’re concerned, Jimmy the Hedgehog didn’t show up this week, because whoever was Jimmy the Hedgehog is now Steve the Echidna.

If it’s unique every time it means they can’t create a consistent fingerprint for you.

A UUID assigned to each user is unique, but that’s not useful for tracking unless you can ensure each user keeps the same number across visits.

If EFF always says your browser has a unique fingerprint then that means the anti-fingerprinting is working, no?