FNG you mention that the Worlds Air Forces do not use Pipers or Cessna's as trainers.
Looking at the USAF they are going back to 'basic' trainers due to the accidents in their Slingsby's, hence their recent decision to scrap over 100 of them rather than 'risk' selling them and being sued for any future accidents with GA owners.
I wonder how many of the USAF fatal accidents were due to the Slingsby dropping the right wing and spinning on the stall?.
Hence their decision to revert to a basic initial training aircraft like their prior to Slingsby aircraft,,,,,C150's!.
The RAF also use 'docile' aircraft, such as the Bulldog, which seems very similar to a Grumman AA5 but with a steerable nosewheel.
I have some 75 hrs in a Slingsby T67, I would NOT have appreciated training in this aircraft due to it's stall characteristics, far better being in a 'docile' staller if it goes wrong when solo training?.
However the T67 is a SUPERB aircraft in the right hands and the USAF trainer was basically a T67 airframe.
Not sure how the Tiger Moth stalls, but surely a point to check before deciding to take PPL in one.