Your check airman is wrong. I regularly flight in out of airports like these, especially in a non-radar environment. This isn't too difficult: VMC or IMC, the aircraft doesn't know there is cumulogranite or airspace conflicts in the way, so you
HAVE to fly the ODP, SID or ATC assigned instructions (which includes ATC authorized visual climb on course)... Either one of those will ensure you don't fly in to cumulogranite or the bordering airspace. Single engine departure off the 30s in SJC has a climbing right turn to a heading of 123 - away from the bay towards the mountains. They don't want your handicapped airplane busting into Moffett or SFO airspace. In the case of DRO, do you want to know why the single engine procedure says to go straight, where as the two engine says turn? Guess what happens to total lift/ drag when in a turn during single engine operations? 50 percent loss of thrust = 80 loss in performance...
I'll say it again because it is worth repeating, your check airman is wrong. Just ask the pilots that turn on course out of places like PSP, TRK, or TVL before completing the ODP and caused a TA/RA situation... Yeah, they got a number to call and had to defend their actions.
You need to re-read 91.175, and call your CFII.
Oh, and the flap position should be covered on the appropriate checklists for de-icing. They're down to avoid jamming or damage when getting sprayed, then up to appropriate setting for take-off. You should also have a flight control sweep after de-icing for the same reason: check for any binding or resistance from ice in the flight control hinges.
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