Former Minutemen
The Minutemen's issues are that they suffered a catastrophic failure in leadership following the death of their last General as well as the loss of the Castle. Then there was the massacre at Quincy which resulted in the Minutemen being extremely demoralized as they not only suffered heavy casualties but they failed in their objective. A lot of Minutemen left the organization because of this event as well as the infighting which severely damaged their credibility (at least in Preston's eyes).
As Preson explains, there's a lot of Minutemen running around the Capital but the organization has collapsed so Preston claiming to be a Minutemen is basically like a guy running around the Wild West claiming to be a member of the Confederacy. Yeah, there's nothing stopping the guy from doing so but it's a meaningless title.
The recovery of the Castle and the various actions which result in a few settlments allying with the "Minutemen" (note the quotes) serve as a rallying cry to the former members spread through the Commonwealth to come and rejoin the reformed organization. Also, it's important to note that due to the extreme hostility of the Post-Apocalyptic World, there's not exactly an absence of soldiers and well-trained survivalists. The Gunners and Talon Company are both fully-trained ARMIES which exist in the Commonwealth and Capital Wasteland.
[censored] armies but armies nonetheless.
We also see retired Minutemen like the former Quartermaster (from "Old Guns") coming out of retirement to lend their operational expertise to the reforming organization. This causes the Minutemen to be something much-much more than just an organization which the Survivor forms from scratch. To draw a comparison, it's like Luke Skywalker reforming the Jedi and discovering fifty or sixty other Jedi coming out of the woodwork to help.
The Minutemen's Ending isn't going to be Great for the Commonwealth a.k.a "The Minutemen are the Desert Rangers"
I mentioned the Minutemen are sacrificing stability for freedom and freedom for stability by being the "middle ground" extreme between the Institute/Brotherhood and Railroad. The simple fact is the Minutemen do not have the technology, the manpower, the equipment, or resources to patrol the Commonwealth the way the Brotherhood of Steel does or the Institute.
They can do a far better job of it than most organizations but they're basically the Desert Rangers from Wasteland and later fit into Fallout continuity via Seth and New Vegas. The Rangers are extremely capable individuals and have a good deal of weaponry at their disposal but in terms of their ability to project power they are in the Minor Leagues compared to the superpowers which the BoS and Institute show.
I liken it to the Independent ending of New Vegas. NCR and the Legion being driven out by the Courier has its upsides as does taking down Mister House but the simple fact is the Courier can't project nearly the same level of authority or social services which NCR can. Nor does the Courier possess the kind of technological know-how which House does (though they can compensate for that with the Big Empty's geniuses).
So, yeah, the Minutemen are the Wasteland ending. Gas mask and trenchcoat guys are great but power armor and android army would be better in terms of pure, "We are not having problems with Super Mutants ANY more."