Coaches, QBs, owners and fans have a strange relationship.
Coaches, for the most part, hitch their wagon to "their" QB and they stay with it. Vick/Reid, Sanchez/Ryan are two good examples. It was obvious, to everyone, that Vick/Sanchez weren't playing good football. I think that's what happened to Shanahan. Griffin is "his guy" and he was going to finish the game unless Griffin said "I can't go." And you know that wasn't going to happen.
The fans either love the starter or love the back-up. Owners want the fans to be happy. If the coach is not wildly successful, and Shanahan has not been in Wash, they want the owner to be happy. RG is everybody's favorite. Until RG laid there like he was dead, that was a one score game. RG wasn't effective but the fans love him and he's Shanahan's guy. I'm pretty sure Shanahan thought two things:
1. We're one play from being tied.
2. If I pull RG and we lose, I'll be 2nd guessed.
Fans aren't happy, owner isn't happy if Cousins fails.