It would be nice if Amtrak could have the "route to GCT" option, but they don't.... nothing at least for a "short-term closure". The route into Grand Central utilizes catenary up to Pelham (a station or so west of New Rochelle), and then its third rail for the remainder. Amtrak doesn't own any electric locomotives that can operate off both catenary and third rail electric. I believe the last ones to have that capability were the NYNH&H's EP-5s of the 1950s, and they didn't last long. Sure, the electric MUs of Metro North's New Haven Line have that capability, but there is no locomotive that can do such today at present.
It is possible that if there were a prolonged closure, it may be possible to run Amtrak into GCT from New Haven, but you'd need the P32AC-DMs (diesel and third rail capability), and those are captive to the Empire Corridor in NYS where they're needed in/out of Penn.
From an Amtrak standpoint, it makes more sense to turn its trains in New Haven if they can't get into NYP. Amtrak has a yard, crews, etc there and trains can be turned there better than at some point on the New Haven Line, whether it be Bridgeport, Stamford, etc. A train can get out of the way easier in New Haven as well, whether it goes into the yard, put on a platform track, or brought up to Amtrak-only territory north of State St.