Sytem reset, before diving into the lengthy read.
The engine restart you mention is only one of the systems reset in the larger picture. Since the EO drill goes deep most pilots think that way. The FCTM explains it properly when read with a naked eye.
Look at your wording,
, when reaching the STS page, the crew should consider any normal C/list, or any additional procedure as applicable
not exactly the true FCTM
is it?
For the "any additional procedure" I never had something particular in mind. Talking to ATC and discussing what request to place with them if obvious might be a good move at that moment. Interrupting the drill in a controlled manner is not a bad choice, helps the brains to catch up.
Your suggestion of APU start, while nowhere mentioned in the books, fits well my understanding of the sequence.
1) you notice the problem, and acknowledge it
2) you fly
3) the 2 of you CRM a little and navigate
4) you check together what actually did happen (FCTM = by observing the OVHD panel too)
5 - ECAM ACTIONS) you secure the malfunction, isolate essential systems etc. This is no solution, just getting the horse back beneath your saddle.
6) you make an attempt to restore what's possible (APU start, ENG relight - well we do not do those actually)
7) - STATUS you learn in detail what the situation is and some consequences:
This makes sense only after having tried to nudge the beast a little.
8) Count your chicken, make the decision to divert, return or continue
9) CRM a little more, three now is better than two before
So at 6) It's engine restart, APU start, ... computer reboot, system reset using the normal control such as stuck PTT. My take it's the moment you want to take a breat before kind of "committing" to the failure as originally displayed.