At one point we were AFSOC, but when deployed we were still owned by the Air Component Commander, hence Half-SOC. I have no idea what would have happened long term had we remained aligned under AFSOC, but it's a moot point as Gen Moseley put us back into ACC almost overnight after a closed door meeting with SOCOM general(s). I've heard all sorts of rumors about what was said, but I have no real data.SOCOM can claim the mission, that doesn't mean ACC would yield the personnel, budget, and airframes. That fight would probably go to the SECDEF level. ACC protested a move to AFSOC a few years ago. There was some arrangement like "AFSOC while deployed, ACC while at home station" or...something odd. I've seen the term "Half-SOC" used to describe the relationship.
Rescue in SOCOM/AFSOC vs ACC comes up every couple years. Personally, I think ACC is the right place, I think the issues amlove has are not a result of ACC but our own internal leadership. That said, we need to embrace the fact that being capable of conducting CSAR (at least from the aircrew side) effectively means that those same skill sets translate to a multitude of missions.
In a shrinking fiscal environment I'd be stupid to say that I only do one niche mission that has a low probability of actually happening, regardless of how important it is. Instead my message needs to be that my primary mission is X, therefore I have the following skill sets: A,B,C, so I can also accomplish missions Y&Z so I will also train to those.