couple things; first, they are making synths as mankind's future. but, "synths are not people".
wait, are they people (set to replace humans), or are they not ? if they are, then the creation of synths makes sense, but so does the railroad trying to free them, and the institute should not try to reprogram synths that escape, rather consider them paragons of the future species.
otherwise, it makes sense to reprogram them, but not to want to destroy the railroad. nor does it make sense to invest resources in hunting down a bunch of synths who think they are normal people. the institute will have gen 4 synths soon, which do not think they are human and do not run away.
there seems to be no reason to destroy the BoS. the institute continues to live underground, nobody can get in, and the only thing which even alludes to the institute exists is the occasional escaped synth (and maybe an abduction or two, which they could do without anyway).
in the end, the institute and shaun are really evil, but without any reason to be. they could literally say "we'll play the long game, ignore everyone else, and emerge in 100 years when everyone else is dead".
shaun says the institute is so well meaning, the synths are created (either to serve as machines, or to become superhumans) to make things better, they have real unadulterated tech + biotech, they are not insane; there is no reason why they could not collaborate with all the factions.
make the BoS into the new military; clear the air about the synths (i.e there were bugs we sorted out, sorry about that) or even stop making synths altogether, get the Minutemen to help you (who have no prejudice against the institute), and explain to the railroad that they are just badly informed. or not, who cares about the railroad, they are a minor faction anyway.
(or, again, ignore them. you have futuristic tech, you don't need any of them)
also...
Kellog abducted your son 60 years ago. he was, let's say, 30yo at the time?
does he look 90yo when you fight him?
all he had was a couple biotech implants, but he was still human.