Well, well. Looks like you've got to the bottom of it. That would explain the ring too.

I was taken in by a virtuoso pigeon artist!

It must have escaped. It was getting a lot of attention at the cafe, but from children. The staff were completely uninterested, presumably because they knew it already. In my experience individual pigeons tend to frequent exactly the same haunts.

It's pride pigeons 😍

It's a game they play here in in the region of Murcia and I know a few that play, trade and buy these "palomos", a top 10 finisher can fetch around 3000€.

Sometimes the pigeons get lost, and their tracker doesn't work to find them, so they can be found in the wild, but it isn't common because they are easy prey for eagles.

Another clue I shamefully missed!

Location mentioned: Teruel, capital of Spanish province of same name. Cafe on main square.

Got a good look, if it was painted it was an amazing job. (PS: it was exactly that.) See added photos. As mentioned, it's not the first I've seen (completely different places).

Yes, I think there may have been a ring, that's definitely a clue I should have been paying more attention to. Too distracted by the fabulous paint job!

Seriously though, I suspect a genetic quirk. Pigeons do have unusual variations due to the human selection in their history. You sometimes see ones with feathered feet for that reason. Like dogs, basically.

PS: looks like your theory is correct. The invisible hand turns out to be very human. I'm impressed.