Louis,
This sounds suspiciously like a CGSC/ACSC or SAMS/SAAS thesis. And this is great, but discussion would be easier if you be a little more up front about it.

Similar questions would include "how is this different from service transportation assets being service run versus USTRANSCOM run?", or "how is this different from the problems the intel community runs into with the 'service missions' versus 'joint missions'?" You might also look at communications, with the conflicts between service doctrines, joint requirements, and systems capabilities.

Are there valid analogies, and are there misleading analogies?

Also, could you more clearly define 'Air Power' for the sake of this discussion?

After glancing at the AFDD 1, I noted that special operations is its own role in "Air Power" operational functions. So when you ask
Does AFSOC require greater power in its role to be able to deliver more Air Power?
I think you may still need to clarify your question.

And (last point for this post, I promise) how often does AFSOC operate without a clear role in a larger USSOCOM operation? I do not have numbers, but I suspect the percentage is very small. If this is the case, it would be hard to justify greater autonomy (where I think you're going with this) for AFSOC, when the relationship with USSOCOM is wired as tightly as it seems to be. In this, your point (as I understand it) is part of a much larger issue of balancing degrees of service autonomy, degrees of fragmentation of DoD, against greater jointness and centralization.

We could go into the culture of the Air Force, and Billy Mitchell's role in shaping the personality of the Air Force today, but that could turn into a serious thread-jacking.