Either way though by the time I was 'trained' on the aircraft both checks
were being done, so whether the teaching method is worthwhile I don't know.
Adding undercarriage to the downwind check early on does help students convert later on, it also adds the extra benefit of primacy when the student is in an extreme situation later in their flying life.
Adding things such as cowl flaps or fuel pumps when there are none in the aircraft is a waste of time. If you land and forgot to open a cowl flap or turn a pump on 99% of the time there will be little consequence. Landing having Forgotten to lower the gear and there is 100% chance of a memorable experience.
The downwind checks such as BUMPFOC and pre takeoff such as TMPFISCH are good when used as a memory aid in addition to a checklist based on the POH. The items could be carried out efficiently from memory and then actions confirmed via a checklist afterwards. The process is a lot faster than having one hand holding a checklist whilst the other wonders around the cockpit. In larger aircraft these basic checks are replaced by cockpit flows etc. You can do the flow checks in training aircraft and it works well however it takes a longer time to learn for each aircraft type. Flows can also lead to errors if jumping between a lot of different types and layouts.
I prefered SUMPFHB, speed (below Vloe and appropriate for circuit), undercarriage (down and locked), mixture (set), pumps (set) primers (locked/off), fuel (mains/fullest tank and quantity verified) harnesses(secure) and brakes checked for pressure last after the gear had extended. All done then do checklist.