Adding up to the OP's question:
1. I've flown for companies that have tailored SOP, as in it's in the Normal SOP section of the FCOM and performed as a flow, from memory); companies that have additional guidance: a QRH section under the SOP tab with the entire supplementary procedure laid down and performed in read-and-do philosophy; and companies using standard, unmodified Airbus manuals.
1a. On this last company, which is the case you specifically ask for, we used to have a sort of shortened version of the SUP procedure in the normal checklist and performed it from there. As a side note, the philosophy across all 3 companies was basically the same: Aircraft on the push? PF handles the controls and starts the first engine. Aircraft under its own power? PM handles controls and starts the second engine.
2. Except for the company with the OET procedure in the Normal SOP section, in all of them, the second engine start was performed as a read-and-do procedure.
3. PF only commands the sequence of actions, i.e., orders the second engine start, the appropriate checklist to perform. No handling of the controls by the PF at any of the 3 companies.
Regarding the age old question of "just how much is worth it", I'll just add a couple of points:
Point A: Any particularly "extreme" of something, in this case application of OET in/out, (i.e, "I never do OET" / "I always do OET") is bound to have limitations, and seldomly one can argument one's such extreme point of view. The many particularities of the job we perform, not to mention the variability of said particularities on a specific day, make this kind of attitudes towards any particular decision hard to justify.
Point B: This procedure has been out there for ages now, and I feel someone, somewhere, should have picked up on something if this wasn't particularly helpful from an economic point of view. Sure, some companies operational environments may not give full benefit from this, as some other company may be greatly benefited from this. Use your head, abide by the limitations and guidelines provided by your company and the OEM and you should be fine.
I've once proposed not performing OET at an airport I usually do, being late at night, no traffic, tired from a long day and on our last sector. As an experienced aviator, we should be able to recognize the opportunities when we can (and should) perform OET. If, for any reason, we cannot, then err on the side of caution.
tdracer While I don't know if you're familiar with the A320, both the CFM56 and the IAE V2500 are equipped on other types you may be familiar with. CFM's seem to be the most rugged ones, needing only a 2 minute warm up under any case. IAE's are a bit more finnicky and do follow what you said: 2 min warm up if shut down less than 2 hours (if memory serves me right, haven't done IAE's in a while), 5 min if over 2 hours. LEAP 1A take anywhere between 1:30" to 2:30" to start (because of the dry crank before every start) and need a 3 min warmup. I'd be interested in reading your comments about any particular engine or otherwise your thoughts about this topic.