It helps the Minutemen are a pre-existing organization which has traditions, artillery, weapons stockpiles, and large numbers of existing trained members who are just "retired" because the organization disintegrated after the Battle of Quincy.
Also, the Settlements were already ready to begin their own NCR-like arrangement when the Institute sabotaged their efforts.
Obviously, I think the Minutemen winning against the Brotherhood makes more sense if they're allied with the Brotherhood and the group is demoralized, scattered, and in retreat after the murder of their dictator in a curbstomp battle.
I also imagine the Laser Weapons and Artillery strikes are the only reason the BoS would be able to be beaten even in a small "revenge" force. Artillery is pretty much MADE for two things, taking down slow-moving aircraft and painted targets after all.