If so, perhaps Stocker could be Starbase 10's "portmaster" as well. But what does not make sense is that Stone indicated he commanded a starship just as Kirk is doing. Stocker, if Kirk's brusque appraisal holds water, has no such command experience. Look at it this way: if Stocker had been at Starbase 11 instead of Stone, would Stocker be able to handle the situation as Stone did?
Stone likely jumped tracks from Command to Support. Much like Kirk probably did when moving to Chief of Starfleet Operations.