In the absence of tugboats, wouldn’t the ship command be given to a local captain?
No, command of the ship belongs to the captain of the ship as per the vessel's owner. Local pilots advise the captain of the ship, but they are never in command.
The United States Coast Guard Captain of the Port (USCG COTP) may set requirements for all vessels under purview to be accompanied by escort tugboats until clear of the FSK Bridge. This would to be to mitigate risk of the terrible incidents like what took place this morning. This may be broken down by displacement tonnage or draft or certain dangerous cargoes or other conditions of visibility which may be a risk hazard along the route. The safety of the bridge and all those that use it and depended on it deserves that much consideration