I've been an advocate of FD being a counter pick instead of PS for quite a long time. A stage with no platforms is more dissimilar to the rest of the group than a stage that has platform transformations. FD also ruins the stage strike balance for match-ups that have a chaingrab on only 1 side (eg, Marth vs Fox/Falco, Sheik vs Many Characters, Peach vs Fox/Falco/Falcon, etc). The characters on the receiving end of the chaingrab are forced to strike FD even if they would normally choose a stage with lots of room, which allows their opponent an advantage in the strike. You can see this at work in M2K vs Leffen from TBH4, they struck to FoD which favors Marth. If stadium was on the list instead of FD, the strike would return to be battlefield centric.
As for Battlefield only for game 1... I'm on the fence. I ask my opponent to agree to the stage most of the time, and even when we strike it usually ends up there. But there is still some player interaction, some mental work, that goes into the stage strike process. I think I would miss that interaction.