The trouble is that IAG isn't a "normal private company" since, certainly as far as Spain is concerned "national pride" is at stake.
Logically you would have a single board or directors, (IAG) with management teams running BA, IB and any other group companies. However you can be sure the Spanish will laregly want to be run by the Spanish, the British by the Irish (!!!!) and whoever else by nationals from their own respective countries.
This sort of logic doesn't seem to run through the former state owned "national carriers".
It will be intersting to see if any of the redundant staff at Iberia will be offered transfers to other group companies - eg. BA. I guess there would be a nationalist hu-ha if Spanish crews found jobs at BA!