[b]Potential Solution:[/b] We don't side with either Zeon or the Feds.

This is just a hypothetical, but what if we took a page from Gundam F91's book and center the story around some ragtag third party? In F91 it was essentially space pirates in the form of the Crossbone Vanguard. Suppose we're playing as something similar? People whose homes (be it a colony or a place on Earth) was hit by war between Zeon and the Feds and were forced into doing whatever it takes to survive in a harsh world/space that's being torn apart by a war. The idea is that we're essentially scavengers. We try to salvage shipwrecks and especially mobile suit parts. We do it to both make money and arm ourselves so that we aren't defenseless. Naturally, neither the Federation nor Zeon take too kindly to people like this potentially learning the secrets of some of their machines and so we'd probably be treated like outlaws or pirates.

In my hypothetical scenario, we have a plausible reason for why no one would start out with a Gundam (because their parts are just that hard to come by for us) but also gives us the potential to acquire one or two of them as time goes on, assuming we survive long enough to reach that point.

[b]Edit:[/b] I also think at least one person ought to play as a masked antagonist of some kind. It just wouldn't be Gundam without a Char clone. Hell, I'll gladly volunteer for that part myself.