I guess the Army just used any old helicopter battalion for their purposes until the Iranian hostage crisis happened, and the same would need to happen for an elite unit to be stood up in the fighter community.
I suggest you read up a little more....”battalion” isn’t even the right term in this case and what you are referring to is Eagle Claw which was Navy helicopters flown by Marines so using “any old helicopter [unit]” shows you have drawn the wrong conclusion....BUT the issues that arose in execution of Eagle Claw highlighted the need for and resulted in creation of standing Joint organization to foster high level of HABITUAL training and successful mission execution. Recommend reading “The Guts to Try” which is the inside story told by one of the lead planners and participants.
I think you need to focus on getting selected and acquainted with the “Fighter Community” as a member (that would be a misnomer for Navy and Marine Corps BTW as Strike Fighter is more appropriate) before you try to suggest a better way of doing business. Regardless of what you call it, the members are pretty aware of each other’s capabilities and if a situation warranted a deployment or dedicated strike mission by a select group of individuals, it would be relatively easy to do within the current Air Wing construct. In fact, it has been done by several different type aircraft when need arose in the past.