If I set the plate at vertical when warm, it's nowhere near closed when cold. To get it to close fully (or almost fully) when cold, you have to rotate the black choke cap to where the mark on it is well past the marks on the housing.
I'm convinced at the moment that the loose shaft is the problem. But of course the proof of the pudding will be in the eating, as the saying originally went.
Let you know if I'm wrong. Should know by the end of the week.