By that same logic, power armor is weak... easily handled with armor piercing ammo... or a strong rope. There isn't much in this world that stops a 50 cal with a thickness measured less than feet..certianly nothing in a vaguely human form-factor. Along the same line, the same mini-gun turned against vertibird means no more vertibird. They're pretty slow moving craft.
And the Minutemen don't need to have infinite range artillery. When you first get it, Preston's advice is to build a network of them to gain coverage of the Commonwealth. To hit the blimp, you just need a few within range of it...easily constructed inside of a junk wooden shed that can be dismantled just before the artillery needs to be used... or constructed out in the open and used to fight mutants while the BoS continues to ignore the Minutemen until that fatal hour when Minutemen turn to drive them out of the Commonwealth in a surprise attack.
Except, of course, there are no fast moving vertibirds. They are slow and flimsy. I'd say they should try for high altitude fat man bombing runs...they'd still suffer heavy losses though, which they can't replace, and the castle is just one of many settlements. Even if they nuke it, the Minutemen are still strong. To win, the BoS would have to nuke all of the Commonwealth, and they have a hard enough time getting vertibirds past bandits with pipe rifles... it wouldn't end well for them.