There are good arguments on both sides

Clean, closer to approach speeds you'll fly in bigger aircraft, especially if you try to fly it fast.

Dirty, have to manage configuration changes, speeds, workload, etc. You'll be doing this in big aircraft too.

Popping out partial or full flaps at MDA/DH, unstabilized approach, can probably work just fine in a slow airplane, but a bad habit for bigger and more complex aircraft, but could be helpful for shorter fields.

I don't really see a "right" answer with what you should do in a lighter aircraft. As said before, they land clean or at partial flaps just fine. If I took students to a class B airport, I'd tell them to fly it clean and keep it fast, that way we'd get the least amount of complaints from ATC, but slowing up and putting in configuration changes, especially in the more complex light aircraft, was good for workload management, especially when flying NP approaches and relating altitude to lose to distance to go. The one thing I didn't like doing was changing configuration upon breaking out, I was always of the mind that you should already be configured and at speed, but chopping to idle at DH because you're at 115kts in a 172 isn't stabilized either...

Especially one of the old 145-150hp broads with 40° of barn door. Hot/high/heavy and it can get a little interesting.

I do find it interesting how the FAA preaches the ‘stabilized below 1000 feet’ bit, but also has ‘forward slip to
landing’ as a testable private pilot maneuver. It’s almost as if they understand that different situations call for different techniques ;)

Anyway, that’s my justification for slipping the nuts off a 172 over the approach lights after flying a 110kt ILS, but I digress. Probably not the best technique for a 141 syllabus :)

The flap motor argument is mute as you would use approach flap and not landing flap until break out.
Also when you break out and continue VFR...well the ‘stabilized approach’ drops down to a recommended 500’
Flap motor? What if you have a vacuum pump failure the moment you go missed? Alternator failure? Or any other single system?
Simple single engine single system aircraft are not suited for hard IFR.
Been there done that, flown approaches down to the ground in 40 year old junk when I was a CFI. Never again thank you.
Approach flap at FAF, if you break out above 500’ AGL you can consider adding flap as required.

Where’s your sense of adventure?!? 😂

Bro.... I could tell you stories about events that have had me waking up screaming at night.
I’m obviously not the smartest as it took me a handful of events to learn from it.
Had an Instrument student lose it and go unusually funky attitudes on me in real IMC in a C172.
To this day don’t know if we rolled It or not.
I lost 1000’ in the recovery.
Stone cold Steve Austin true story.

Lol I believe it.

My opinion for flying a trainer is as followers. First of all, the requirement to fly a stabilized approach does not apply to part 91 (I know you're at part 141) unless of course it is company SOP. So, while it is a solid concept, let us remember that. Now if I were flying a 172 in Class B or C or, heck, even D airspace, it is probably safe to assume that there is more than enough runway to land flaps up and keep your speed up on the approach so as not to interfere with heavier traffic too much. If you are flying into a smaller airport where flaps are required to make the landing distance, I would configure prior to the FAF or perhaps a couple miles out from the DH or MAP. Chances are you won't be holding up big traffic in this case or at least not dozens of aircraft. Now I do see some training value in changing flap configuration after breaking out (perhaps teach in VMC if you're nervous). The student should be able to react to the change in attitude and adjust power and pitch settings as necessary. We should be teaching pilots their stick and rudder skills in the little prop jobs so as to set a solid foundation for the rest of their careers. Honestly, in a 172, I don't think that it is too unsafe to change the flap setting at breakout. Your flight school is probably trying to save money on fuel too. It burns a lot more gas to have the drag makers out with 5 or 6 miles to go.