I think the inherent issues of using Gen 3 Synths as a subservient race combined with the Institute’s proclivities for research that is morally ambiguous at best and outright evil at worst would still drive a wedge between the core of the Institute and the inhabitants of the wastes.
Looking specifically at the people of the Commonwealth, they do not hate the Synths because of how they are created or for the same ideological reasons as the BoS. For the average waster, their hatred for Synths comes from years of the Institute being a boogieman, sprinking out gen1s and gen2s as shock troops to inforce their will across the surface. Or worse, the constant very real threat that at any given time, any one you know or love could have been replaced with a duplicate; that *you* too could one day be face to face with a your own Institute brand replacement.
In what-if, there’s likely no reason those existing fears would exist. So long as the Institute is a resource and a benefit, it is very likely there would be a positive association with synths. However, I do believe in the absence of fear or hatred for synths, that in the case of gen3s, there would be a great deal of compassion or empathy for the synths. With the early generation version of synths, there’s still a substantial barrier between technology and humanity. A gen1 may as well be a protectron with more points of articulation. A gen 2 may as well be the male counterpart to the female assaultrons. It’s a clear usage of technology that operates in manner similar any Robco creation.
Gen3s, however, come with all the “are they people” baggage that is tossed around every single time we see the discussion about Gen3s pop up here on the wiki. And without opening that specific can of worms, there’s no doubt that the people of the wastes would have some of the conclusions. The railroad is already motivated by these feelings in F3 and F4, and in F4, we do see that there is a small but present element of pro-railroad sympathy from within the Institute. In a what-if where the common man is exposed to the day to day mundanity Gen3s in an environment outside the sterile walls of the Institute proper, I believe it would be far easy for sympathy to grow.
At the very least factions like the Railroad would be just as present in this what-if world as they are in the Commonwealth we know and love. Likely the BoS would be unaffected by the state of Commonwealth politics unless a series of events unfolded in a way as to prevent the gen3 from coming into full production.