I'd have to disagree about "I don't want examining". It's fine, at least in AustE and BrE. Some other examples of this kind of usage:
"The car's fine. It doesn't want fixing." (= It doesn't need to be fixed).
"That picture wants straightening up" (= It needs to be straightened.)
"That guy's a mess. He wants a haircut and some new clothes." (= He needs a haircut and new clothes.)
Thus "I don't want examining" is perfectly correct as a way of saying "I don't need to be examined", though perhaps not in AmE.
Right, I don’t think it’s correct in US English, but based on your post, it sounds like even in your variety “want” means “need” there, so it still doesn’t work as a substitute for “I don’t want to be examined,” does it?