I've finally destroyed the Institute in my first run through, assaulting it with Liberty Prime and Maxon and the rest of the merry band of bucketheads.
Now, I realize that this requires a significant amount of Head-canon, but I feel that this ending allowed me to have my cake and eat it too.
First, my assumption is that the BOS absorbed some remnants of the Enclave, scientists and grunts and whatnot, upon the destruction of the Crawler. This, along with the absorption of the Outcasts, is what allowed them to build such a powerful and industrialized nation state capable of force projection (though that's probably not the right word) in the capital wasteland so quickly. Secondly, the BOS has a policy of accepting wastelander recruits.
Secondly, the heart of the East Coast Brotherhood of Steel is and remains the citadel in DC. The Prydwyn is a mobile command center, sure, but it doesn't have the industrial facilities necessary for the obvious industrialized might of the BOS (Vertibirds and T-60s, for one.) This, along with the habit of the west coast BOS of scattering chapters every-which where (Chicago, Colorado, Mojave, Washington DC) suggests to me that a Commonwealth BOS chapter is Maxon's Plan after the destruction of the Institute, while Maxon and the Prydwyn itself returns to DC for eventual operations in other places. (The Sunshine Wasteland? FO5?) Since Maxon names me Sentinel at the end, and gives me discretion to conduct my own operations, this suggests that I would be made head of this New Chapter, reporting to Maxon, but functionally independent as the distances between Boston and DC are pretty huge.
Thirdly, during the assault on the institute, I activated the evacuation code. While some scientists were killed, the crowds that were present in the Institute in earlier segments were absent. This leads me to believe that many, if not most, of the institute's inhabitants evacuated. This allows for many of them to disperse to the various settlements AND to get recruited and absorbed into the BOS itself, as the BOS once did to the Enclave.
Finally, I am the general of the Minutemen. I command them, and have earned their respect many times over (Goddamn repeating missions). As General AND Sentinel/Potential Elder of the BOS, I can bring those two organizations together and recruit those minutemen that want to serve full time in a defense force for the commonwealth as aspirants, knights, and eventually paladins of the Commonwealth BOS, while letting those who want to farm, farm. This willingly brings the network of settlements and farms and minutement that I've built into an established chain of command, preventing the re-collapse of the commonwealth into anarchy.
This put together presents a future for the commonwealth that is a hybrid of the BOS and Minutemen endings, where a semi-feudal contract exists between the Farmers and settlements and the BOS who protect them. The BOS, from their fortresses of the Airport (which needs a more badass name) and the Castle, which now serves as command hub for the Commonwealth BOS, and having absorbed many of the Institute's refugees, brings Peace, security, and prosperity to the Commonwealth. the Institute's scientists supplement the technical knowledge of the BOS with their knowledge, all while trying not to make too many waves, and their children would be raised as good commonwealthers and BOS Scribes.
Maybe I'm idealistic, but, well, this is absolutely my head canon.