So do you know why they don't use id tech for all their titles? I mean maybe apart from mobile, it would seem like a sound choice and can be akin to EA with Frosbite, so that the engine is versatile for all developers, and thus they can share all their techniques, infos, debugs, etc. Does that not make sense or did they start development already on the engines you mentioned above?
idtech was not really an appropriate choice when most of these games started development. It had major limitations and required pretty small play areas, along with having astronomical file size ballooning that would be problematic in games with a lot of assets.
EA has 200 people working on Frostbite to make that proposal work.