How to balance the requirements between shooters and support ships is up pto fleet staffs but how to recover personnel is something that would need to be considered. Keep in mind that we're talking about PR as though it only applies to Aviators. In a big fight there will likely be a need to recover Sailors from ships that have been sunk by enemy action. The USN maintained a strong PR effort in WWII and it paid dividends in terms of morale and in returning experienced personnel to the fight. The IJN didn't do the same and lost many experienced personnel and left them with very few experienced personnel by the end of the war.
@BigRed389: concur that picket ships near an enemy coast would require maritime supremacy as I mentioned in an earlier post. Or a willingness to take losses to support the big picture such as at Oki when the radar Pickett DDs suffered terribly from kamikazes to protect the CVs and landing force.
I understand why you wouldn't want to tie up a DDG doing lifeguard duty but there may be other ways. Maybe a use for LCS? Or keep a long term PR det afloat on an LPD or other ship that may be less used in a peer naval conflict. Or just use the DDG as a lilypad for PR assets on the CVN. I'm in no position to know what's best, just some need on the internets.