No one has mentioned the other problem with paper or similar check lists in the circuit. It's congested in some cases, so the plan is to be looking out of the window as much as possible to make damn sure that there's not about to be an encounter of the wrong kind. Messing about finding the right page on a check list, then going head in the office to read it is distracting from the main requirement when close to an airfield and or other aircraft.
To me, the best downwind check is the one that covers all the options, so undercarriage, mixture props fuel and other issues need to be in there, that way, when a new maybe faster multi engine type is being flown, the basic check is identical, and still covers all the bases.
If the CFi has a problem with that, then maybe someone needs to (gently) try and find out what the CFI's problem is.