From the perspective of settlers, most of them came in with nothing and the Minutement constructed farms for them to tend, pumps and purifiers for clean water, power for lights and music, and defenses against the horrors of the wastes. The BoS mainly just crash vertibirds around, probably killing a few settlers as collateral damage... while demanding food and just mowing down anyone in the way. Well.. mowing is a vast overstatement. More likely the settlers were all 'Oh those super mutants over there are bad news... stay away!'....and then the BoS Knight was all 'Nah mang, I got his' before rushing in and getting murdered by a swarm of super mutants with heavy weapons. I think the settlers would see the BoS as unpredictable and dangerous. Meanwhile, the Minutemen seem to be their only hope at a good life. The Diamond City radio broadcasts seem to reflect that sentiment. The DJ is afraid of getting murdered by the BoS, while he acknowledges that whatever is good for the Minutemen is good for everyone.
When it comes down to the wire, the people are going to throw their lot in with the group that has given them a place to call home.
I really wish the game had story arc that let you take control over the Commonwealth BoS.. perhaps leading them to separate from the main BoS and claim/rebuild the Commonwealth. I love the BoS..or rather what they were. Now they're just garbage.
I don't know about you... but I equip my Minutemen. If you give them good armor and proper weapons, they can slaughter about anything. I doubt BoS power armor would even pose much of a challenge.... i might have to spawn in some knights and give it a test run. Also, power armor apparently requires very little training. You can leave it around and even a lowly settler with a pipe pistol can jump in and use it to fight.
On the other hand, I'm beginning to suspect that the BoS presence in the Commonwealth was a setup from back home to get rid of an ineffective leader. I'm not sure how else to explain the how terribly ill-conceived this whole expedition is.