'CinC' Commander in Chief...normally only applies to the President. In Star Trek that may be different.
Commander Starfleet, Starfleet Commander whatever, looks to be the title for the head of Starfleet that's been used on screen.
'Chief of Staff' is (in the real world) mostly an administrative assistant position and not in the direct chain-of-command....ie: the Starfleet Commander would have a 'Chief of Staff' to assist in the daily running of the commander's office....the CoS would not take over if the Commander was incapacitated...that would fall to the Deputy Starfleet Commander (if there is one) or the next designated senior officer. (Probably Chief of Starfleet Operations)
And don't confuse 'chief of staff' with the title Commander of the Joint Chiefs of Staff that the US military has...whole different ball of wax.
In the Star Trek universe its been pretty well established that there is an over all 'Commander' in charge of Starfleet...followed by chief's of other departments...Chief of Starfleet Operations, Chief of Starfleet Logistics, Chief of Starfleet Personnel, so on and so forth.
Wasn't Kirk at the time of TMP listed as 'Head of Starfleet Operation/Chief of SF Operations'?...and he still had to go 'upstairs' to talk Nogura into giving him the Enterprise back. Indicating that 'Chief of operations was a subordinate position to the Starfleet Commander.