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
That’s actually not totally true
Pilots think they are actually in command and typically capt gives them the helm
They are not simply advising
They can be overruled and argue
I’ve seen it with my own eyes
And been on radio directing my capt to take the helm minutes before a scrape with downstream Chem carrier while we were upstream for comet American rice on the calcasieu at lake Charles
One thing pilots will do is immediately disclaim any responsibility when they do screw up
Bar pilots I’m more fond of than river or harbor pilots
All are overpriced in the USA especially lower Miss river ports
But if there were tugs present where we’re they?