I guess that would be doable technically, but that strikes me as extremely counter-intuitive and immersion-breaking. Like the choice of species, this is something done "out of universe" (as something choosen before the game starts), while all the decision in-game are more or less "in-universe" decisions.
So, while we surely can make it possible to choose you're origin after game start, that feels bizarre - I mean, as a race I can't venture into space, look around and then decide what my origins, my past have been.
Anyway, I think you're worrying too much about how the choice of Origin would predetermine too much which could come into conflict with the situation you find yourself in at game start. Actually that mechanic wouldn't add really anything new - we currently already have that concept in the game, in a way. It's just baked into the species definition right now.
The main thing Origins are supposed to accomplish is to separate things that don't need to be species specific into a second set of choices. First, to give the player more flexibility at game setup, and second to better honor FO's species != empire philosophy.
In essence, your choice of Origin should not have a worse impact, regarding predetermination of things too much, as choosing your starting species already has.
I mean, we specifically designed a species (Sly) for the hidden empire gameplay. Decoupling such a choice from the species and creating a separate game setup option for that doesn't change that much, it basically just makes it possible (in this example) to enable the hidden empire gameplay for all species.
I get the feeling that we are (or will be) getting more "specialized" gameplay options (like the hidden empire one) down the road, and I'm not very fond of the idea to have to create a special species for all of them in order for players to be able to make use of these options, and these options only being viable when choosing the right species for it.
I want to have a hidden human empire too