The student flying on an instructors licence thing is an urban myth - they fly under the ANO Article 52 exemption, which as discussed does need to be as directed by an FI not a CRI.
I have very mixed feelings about CRIs, I know some very experienced and capable ones but have also flown with several who wouldn't even pass a PPL skills test and whose instructional ability was frankly dire. I think a lot of it comes down to how much they do / how current they are and how they are mentored.