Hmm, maybe they sent out the robots they had to collect resources?
And when coming into contact with the outside worlds people they enslaved them to work in mines and stuff to collect more resources.
One of the reasons for the synthetics could be to replace the slaves.
That's possible, but it woudl still be quite an effort to start really progressing without a working societys help in manufacturing certain parts and materials that just don't lay around ruins or that could be mined. Nice idea though, a somewhat cyberpunkish feel to it which I like.
But, I don't particularly like the thought of androids/synthetics in Fallout because it kinda goes against premise of the tech the universe evolved to. Vacuum tube computers, clunky and heavy robotors, and no any particular miniaturisation or microchips. I think that the idea with the institution should be kept on a leash to not overdo it, keep it simple and with a premise of pointing towards a possible future instead of a certain one. Depict a "design in progress" scene with a sense of uncertainty, so to speak, instead of "mass production already ongoing". Something like that would be my preference if the idea of androids is expanded upon (and if it is, there needs to be an electric sheep somewhere).