I don't see Boeing pilots learning Air Bus drills just in case one day they might fly an Airbus.
You've never worked for an airline flying both types then - it is REALLY common for one type's checks to bleed into the other for "airline commonality".
As to generic checklists - you have one endorsement for single engine less than 5700kg. The school knows that your licence means you can fly any of these types, so teaching a generic "one check fits all" system makes sense.
Sure, if you're flying only one aircraft, feel free to use a manufacturer's POH type-specific checklist, but I know I was happy to use the same memorised checks for warrior, C150, C210, Navajo, Baron etc etc.
... and I was happy to teach the same as an instructor for that very reason.