The locomotives with "a shoe that rides the third rail" are the P-32ACDM class;Road numbers 700-717. They use the "Genesis" carbody in common with the P-42 class.
The P-32ACDM are assigned to Albany, and regularly call there in assigned service. There are no problems positioning those locomotives for periodic maintenance.
The New York commuter agency, Metro North, also operates this type of locomotive as well. These locomotives of course assignmed to Metro North trains over the NEC between New Rochelle and New Haven.
But, barring some kind of emergency, I cannot find reason for an Amtrak P-32ACDM to operate over any portion of the NEC.