So I'm definitely not hearing anyone say that Club Baggage is supposed to have actual priority handling (still less that it works in practice).

The good news, I suppose, is that it means I wasn't imagining things.

But it's disappointing: I assumed that the modern baggage systems would be able to put such a priority into practice.

I just had a look on ba.com and I didn't find any claim of priority handling for club class bags. So perhaps that has never been the case ?