Do we know for certain that SHIELD is no longer getting funding from the Security Council/presumably the United Nations? Like, if the FBI was infiltrated by, I don't know, the Mafia or something, and the Mafia managed to kill hundreds of FBI agents and cripple most of their operations before getting found out, would the US government just say "Well I guess the FBI is over now," or would they keep funding what remains of the organization in order to rebuild?
I'd think it would be the later.
Anyway, as for Coulson's promotion, he may very well be the highest ranking SHIELD agent who isn't dead, in prison, or on the run from the law right now, so his promotion might have been automatic from a chain of command standpoint and Fury was just letting him know that/pushing him into the role.