It's been a few years since I did instructing, or flew a single engine piston for that matter. Despite, I'll try to share my thoughts.
(1) From the very first week in flight training, even before we actually practiced any form of emergency, we were required to know the engine failure checklist by heart. Following an 'engine failure' we would get it done, touch drilled, within seconds (Fuel selector, Fuel SOV, Mixture, Aux Fuel Pump, Ignition..as for the trouble shooting). Similar to the memory items in an airliner. As I recall we never even picked up the checklist, which resulted in a crude awakening in FAA land where it would be a fail item for not using the checklist. To my knowledge, I never got it wrong though..
We all know it should be weighed against hight (or time) available. This decision - to fly the plane first - will be done close to split second and, for an inexperienced PPL holder, more conservatively than an experienced instructor. I have seen the polar opposite too of those who tried to get the checklist done no matter what, when the situation clearly asked for someone to fly the plane. Could it be that they were too unsure of the checklist content (or memory items if you will) that in the stress they just proceeded to focus on getting the landing done safely?
I always taught the engine failure drill as memory items and only use the checklist if time permits. Sending an emergency message and briefing the passengers I see more important than pulling out the checklist which, if you've studied well, you'll get right 99% of the time anyway. But this might not be how the real world looks like with some only flying a few hours a year?
(2) Dynamic situation with no clear answer, in my opinion. As you've witnessed there are situations of field fixation. I remember being subject to it myself. I think, however, there's also merit to sticking to the plan. I have seen some acts of late decisions leading to a right messy situation. Much depends on the terrain. If there's miles of flat land a late decision to change is more forgiving than if it's forest or rock. As an instructor I think it's our task to find that balance. Tell your student when they missed a perfect opportunity down on their right side, but don't make it black or white. It will widen their perspective. Again, experience level and stress will affect when a field swap is appropriate and when not.
(3) Is DODAR taught also in the context of flying single engine piston? This is just my opinion, take it for what it's worth. Even in an airliner context this mnemonics are a bit overkill. They look good on paper and in CRM classes but is rarely of any real assistance in the flight deck. I was one criticised by a trainer for keeping a 150 nm fix ring of a suitable alternate for time critical emergencies (fire or fumes, heart attack etc.) with the motivation I was suffering from confirmation bias. Instead I should run a decision making process a la DODAR. If anything they are for non time critical abnormal situations as an assistance to channelise our thought process. This is something we rarely practice in single engine private flying, but maybe we should? An example would be you're on a long cross country flight and you recieve reports that the weather is closing in on your destination.
Regards,