1) I brief then demonstrate how to fly the aeroplane, and if required include a brief on the airfield and local area.
(2) Checkee (is that a word?) tries to fly it, with formative feedback from me.
(3) Checkee demonstrates hopefully that they can manage on their own whilst I sit there and say nothing meaningful about the flying.
(4) Either (a) welcome them to the syndicate, or (b) politely encourage them to go and get some proper instruction.
I have been through many check outs. I would disagree with most of what you've said here.
First, you don't brief the person being checked out, they should be briefing you, if not, then they don't get in the plane, cuz they don't know enough about it to be able to fly it.
Second, you don't show them anything about the plane, but rather, like an uneducated passenger you ask annoying questions about how this or that works, and have them show you.
You are checking them out, not the other way around.