I kind of agree that it shouldn't be faction specific but a DLC that potentially expands on all the major factions would be interesting. Especially if there is the potential to bring in some of the older enemies, such as the ones form the pitt that you mentioned. Just the implementation of a group trying to conquer the commonwealth could be and interesting DLC, not unlike broken steel in FO3. Realistically it seems like you could expand on any one of the major factions' story line with the first idea, not just the minutemen.
Institute - Raiders try and take over and you have to secure Institute settlements and the remainder of the commonwealth essentially falls to the new raiders save for major locations. These major locations could be taken over Skyrim style.
Minutemen - You must unite the commonwealth to ensure that all colonies/cities maintain independence, not unlike the revolutionary war.
Railroad - (Raiders could be enslaving synths and/or settlers) Ensure that the standing railroad is not jeopardized by securing settlements and opening them for the railroad and using it to liberate slaves that the new raiders are taking. This could additionally open up the option to become a slaver and help take over the commonwealth
BOS - You must use this opportunity to strike against the various factions to ensure that pre-war technology falls into the right hands. Again Skyrim style city take overs, etc.