I'm just curious since the worry about 'internet connection' issues and fans not cooling, when they're initially powered up is that any concern?
not a concern re overheating initially as it would take it a minute or so to situate itself on your network and then hashing will build up slowly (don't expect it to jump to to 4.86 in few seconds). Wait at least 1 hr before reaching any conclusion where your miner is operating as far as speed is concerned and if high errors is seen, try soft reboot in gui (I believe that it is in system pane).
I think that you are precisely correct when you say, "turning them on one at a time with the control board psu being the last one to turn on." I have quite a few S7s now and have several powered by 3 power supplies and followed the power on procedure that you noted without any problems.
How many watt PSU per board do you use - all 3 the same or more on the controller board?