And basically thats exactly what happened.
The in-game screen transitions mention that the Institute used to work more closely with the people of the Commonwealth, but cut off contact after mistrust was sowed between the two for various reasons.
And in addition, the reason the Institute scientists needed Shaun, was because most of the Institute's DNA had been altered by surface radiation. Suggesting that at some point in the past, the Institute worked regularly above ground, which is whats causing the damaged genes to be passed down to the current generation of Institute scientists.
So yeah, the explanation is already there. You just need to pay attention to details.