You have this great "Leave No Man Behind" policy, which is really admirable. The same policy that means the likes of John McCain didn't die in a Viet Cong prison 40 years ago. Prisoner exchanges are also a pretty standard diplomatic procedure.
Now as for the accusation of Bergdhal being a traitor and a deserter, that will all be dealt with in good time. You can be pretty sure that the moment he arrives back in the USA he'll have a dozen different organisations interrogating him over various facts. The DoD already has extensive files on everything already. They really just need his testimony before starting whatever kind of court-martial or other proceedings against him. Ultimately it's military courts that will decide what to do with Bergdhal, not Fox, not CNN, not MSNBC or any other interested media party.