I'm not sure what "priority baggage handling" is all about. I know that sometimes flying first there is a different tag put on the handle. But I have rarely found that the bags with the first class tag are handled any differently.

I often travel with samples and inventory which requires three pieces of luggage. The policy regarding bags that is very clear in print, but is little known behind most United and Star Alliance counters, is that Star Gold cardholders are to be allowed three items of luggage. This is in print everywhere, but I have been told sternly by several agents at check-in that I am going to be charged $50 or more for excess baggage. I then respond just as firmly and clearly that no such thing will take place because of the published, longstanding policy. And then I hand them a printed item that I carry with me everywhere (I think it's a page from the letter that comes with our statements), for this purpose, that says clearly that Premier Exec/Star Alliance Gold is to be accorded three checked bags. Having the printed policy in hand seems to do the trick. They can find the policy somewhere deep in the archives on their computer screens. But nothing seems to work like a dated piece of published policy.

I've had such incidents in Istanbul, Tokyo, and about a dozen places in the US.

That's the only bit of luggage privilege that I've experienced on Star Alliance. I've learned that I have to be ready to be the enforcer myself.