i finished the game and i nuked the institute with the brotherhood of steel.
but i do feel sorry for killing them all it's just i didn't want to lose my rank in the brotherhood.
Elder maxson is a madman
i reckon the brotherhood is evil not entirely it's just maxson and his delusional self of being a demi-god
The Institute is a slave-state supported by Synth Labor and would collapse without it, isolationist, have been performing human experimentation on Wastelanders, and then there's the whole thing they've done the "CIA props up dictators and destroys democracies" thing in the Commonwealth but they add the benefit of using "Invasion of the Body Snatchers" robot clones of people, basically.
OTOH, the Institute is the single greatest repository of Pre-War knowledge you'll probably find and wants to make YOU the leader of it. If you destroy their enemies, they think they've had enough academics and need someone with a bit more substance. Since there's no ending slides, you could easily headcanon your Sole Survivor liberates the Synths gradually (or replaces them with dummy robots) and helps the Commonwealth become an NCR-like state with all the vast technology of the Institute behind them. Or, hell, could use Generation 1 Synths to rebuild the Commonwealth into a technologically advanced society with none of its current monster problems.
The Brotherhood of Steel's biggest problem is that it's basically run by a guy with delusions of being a God and is an enormous genocidal racist. Elder Maxson intends to murder every single nonhuman in the Wastelands and has convinced his followers Elder Lyons was tired and ineffectual (which he probably was but he had the LW to prop him up). There are decent ghouls and Synths by the hundred in the Commonwealth and they'll all die if the BoS wins.
OTOH, omlette eggs. The BoS ruling the Commonwealth AND Capital Wasteland would go a long way to rebuilding the East Coast.