@pinteam
According to Airbus the check should be done with a new crew. How closely each individual airline follows and why (not) would be an endless debate, though still enlightening at some points.
AFAIK running the APU is a not part of the DY.
The quoted sentence "If the APU is already running, ensure that the following check has already been completed. If not, perform it." is
not in my recent FCOMs but I see it in the 2014 edition.
For your last sentence: crowd mentality and assuming things can be a biting bitch. For instance, I hear of an airline where the agreed saving technique was to shoot approaches with F3 and increase Vapp by +15 kts for "safety" and "easier landing".
Or another one, where when calculated ASD=ASDA on flex takeoff they would modify runway by -100 m at DER to get a "margin" - of course only on the long runways where it was possible. Emergency torches being used for walkarounds "to test them" as supposedly they re-charge when back in the cradle. Having already left a cloud into the blue, E-AI must not be switched off until ECAM MEMO : ice not det, in order to "prevent getting a QAR".
If your OPS always start in the morning at a base where MX power-up and prepare the A/C, including running the APU, there is probably no need to change the drill. Situation-driven SOP, nothing wrong with that. The book provides protocol-driven SOP that are situation independent.