The BoS in fallout 4 are basically the start of an autocratic empire. They go where they please and do what they please in the name of their own goals and remove any obstacles in their way because in their mind, might makes right.
Sure, they try to justify it and rationalize it but in the end, they're bullies, plain and simple. They would stabilize the Commonwealth, that much is certain, but they would turn its residents into lackeys who have the option of either cooperating or being killed off as a "potential threat to the stability of the Commonwealth". All the while the BoS is claiming all advanced technology for themselves under the pretense of "keeping it from being misused". The best way to prevent misuse is education and careful monitoring, not taking things away.
Oh and what makes people so sure they'd stabilize the region? On the Prydwen, you can access a computer which contains the letters soldiers wrote home (i.e. the Capital Wasteland). In one of them, a soldier orders his loved ones to pretty much sleep with a gun beside them in case of trouble. Does not seem all that stable to me.
But anyway, the BoS would quite literally keep the Commonwealth stupid and terrified of their might under the pretense of keeping it "stable". Textbook autocratic regime stuff.
Personally I was surprised they nuked the Institute instead of at least nicking all the tech beforehand. I think you can look at this in two ways, first being that Bethesda didn't really commit to making the BoS seem like an autocratic regime and the second being that the BoS are buying their own propaganda so much they started to believe it themselves.
Likewise, the Institute would also like to keep the institute stupid, paranoid and terrified under the pretense of doing "the best" for them. Sure, they might make it a blossoming pre-war paradise in a couple centuries, but that's a maybe. The commonwealth's population would certainly not appreciate it and neither would the Synth (who are lest we forget manufactured human slaves).
The minutemen, on the other hand are a very reactionary force, that much is true. But as we know from history, first kingdoms and nation states have been born out of the need for protection which brought multiple settlements together.
The minutemen are precisely that, only with far more autonomy. Under the proper leadership (a charismatic, strong and determined leader such as for example the Sole Survivor when they are doing the Minutemen quests) they could unite the Commonwealth into a formidable, stable nation.
And even if they don't, they are a strong peacekeeping force that could easily take on the raiders and super mutants of the Commonwealth and win.
especially since with the Institute gone, the population of Super Mutants has no way of recovering.
The biggest problems would be dealing with the Gunners and Children of the Atom.
Now Gunners are a very typical Bethesdaism. They are a highly organized, well equipped faction who are there to give the player a meaty mid-to-late game challenge, but make no bloody sense from a worldbuilding perspective. They are clearly intelligent and organized, unlike your typical raiders. Yet they have no discernible goal or objective. What they do is build highly fortified and well defended encampments in the middle of nowhere and then proceed to sit inside them and do bugger-all except take potshots at anyone passing by. If they wanted the money, they should run protection rackets, offer really expensive safe havens for merchants and do mercenary work for the settlements now that the Minutemen aren't around to do it out of the kindness of their heart. But no, they just sit there in their fortresses attacking random people and generally making no bloody sense.
The minutemen would be in for quite a fight to take them on and it'd probably take several bloody, exhaustive campaigns. Or they could do it diplomatically (in a reasonable world, they would but this is a Bethesda world so they most likely won't).
Now the Children of Atom are a much smaller group but a very significant threat. They are equipped with powerful weaponry (the Gamma Gun will wreck your shit unless you've got some serious radiation protection) and are on a religious crusade to purge the unbelievers from the Commonwealth. Again, a tough fight for the Minutemen.
So, the Minutemen could potentially stabilize the Commonwealth, but it'd take them longer than the Brotherhood (even longer still if they'd pissed the BoS off and had to take them on as well). However unlike the Brotherhood, they do not seem hell bent on stripping the commonwealth of its technology and then keeping it "stable" by brutally crushing anyone who dares oppose them. However, their overall behaviour seems to be very dependent on the whims of their current leader. Then again, so does the Brotherhood's.