If you're still following it, apparently Hinch never even sent Cole to the bullpen or asked him to warm up. He wanted his relievers. Most of the criticism I've seen of him has been that he pulled Greinke too early but I think that's pretty easy to defend, even without considering recent track record. Maybe he gets the same result Harris did, maybe he doesn't. At the end of the day that's where your high-leverage reliever makes his money but they are also not perfect - nearly every close playoff game is lost by whichever teams wins that battle (including most of our big ones in '08 and '09).
Nats ability to get meaningful relief innings out of Corbin a really underrated factor in the series though. Astros had similiar success in 2017 when they'd essentially lost Giles.
Anyway I think the post-game Cole drama was because of that, he was pissed he wasn't asked to pitch but Hinch was probably pretty peeved his star player on short rest warmed up before being asked to (especially since his stated intent was to save him for the 9th).