That’s an interesting question. I suppose the first thing to respond to is this:
Just because the BOS as a faction wouldn’t exist doesn’t mean Power Armor wouldn’t exist.
Generally speaking though, if the BOS didn’t exist, there would be a lot more dangerous gear lying around the world for people to use. There would be people, good and bad, who would most likely wear a suit of Power Armor such as the Hub guards or an even scarier thought, the Regulators of Adytum. Gun Runners would probably be the next best bet after the BOS to wear and upgrade Power Armor and perhaps laser pistols through schematics that would be acquirable without the BOS’ hoarding.
If I’m not mistaken, the crack assault team of Paladins is only for the Mariposa Vats and not for the Master himself, so little changes in that regard. I’d say the Vault Dweller still succeeds in taking out the base either by Infiltration with Children of the Cathedral robes or a squad of Hub police who I think could be convinced to help if they had Power Armor and knew of the oncoming invasion (which you reveal to them when you do the missing caravan quest).