Okay, these are the facts:
BoS:
- population (I have not confirmed this from anywjere) I could not imagine over 10000
- broken liberty prime, maybe some other super weapons
- separated in multiple groups
- power armor
- advanced technology
- don't produce anything (not sure about this)
- 1-10 vertibirds (broken steel)
- mosty trains/recruits only "inside brotherhood" born, humans
Wrong about a couple of things... Population of the Western BoS unknown they don't let in outsiders. Midwestern BoS let in outsiders, mutants and have an army of Robots, they span most of the midwestern united states. Eastern BoS, they let in Outsiders. So Midwestern BoS could have well over 10,000 (they don't all have Power Armour) and Eastern BoS might have a number close to that.
They do produce things, fallout one and three show scribs and their job is to figure out how things work so they can repair and make new things. We have not seen mass production so that could be up to debate but I think it odd they go through the trouble of learning oh it works without making their own.
Western BoS could have unlimited Vertibirds due to the quest to steal the blue prints from the Enclave. Wanting the blue prints is another sign they have the ability to make their own, only if they knew how to make one (blue prints)
All these are because of the good endings to the game and in fallout the good ending is always the Canon ending.