I agree with you in that I too think there is something not quite right about the way FoG does this at the moment, I just think your solution might actually be worse
Having more mounted does not IMO make you more likely to be able to choose whether you are invading Italy, or waiting for me to invade Mesopotamia, hence terrain type choice should not be influenced by tactical factors like number of mounted. Or quality of CinC either.
And if we are manouevring around e.g. in a steppe region, even if you tactically manouevre better than me we are still in a steppe region, and that's where we will fight the battle. Who gets a good defensive position of their choice is another matter - if I'm Crassus I try to make my way to those small hills I can see in the distance, whilst your Parthian mounted chaps try to bring me to battle in the open before I get there - so *exactly where we fight the battle is to be decided somehow, but it's still a steppe region.
I personally like the terrain choice and placement system in the *cough* Other Ruleset (TM) better, although I'm not suggesting FoG should do exactly the same! (Especially since I prefer the FoG system for deployment of units).
So...decoupling terrain type choice from the PBI roll would get my vote. Maybe just make it a coin flip as to who chooses terrain type, to avoid adding extra complexity to rules or army lists? The dice rolls when placing individual terrain pieces introduce randomness that abstracts the pre-battle manouevring for a good position - if the dice allow me to put terrain pieces where I would ideally want them, then obviously my pre-battle manouevring was effective.
Then the advantage of winning the initiative roll would be siomply to have the choice of deploy second, or move second.