I can't find the quote you reference. Do you have a link, because I have a hard time believing any player or coach wearing Burgundy & Gold even alluded to it. I listened to a week and a half worth of podcasts and regional radio shows leading up to and following the Detroit win, and leading up to Philly. According to Craig Hoffman, the message in the locker room was, "The road [to the Super Bowl] goes through us."
The Eagles didn't want it more. The Eagles were the more complete team and they were the better team on Sunday. They made less mistakes, but Washington won 12 games in the regular season. Then beat the team that blew them out 18 weeks earlier and the #1 Seed, both on the road. There is no way they believed they didn't belong. Washington scored 22 points in the 4th quarter of their last meeting against the same defense, so they were still in the game until the Eckler fumble. 9 points at the time was not, by any means, insurmountable.
Dan Quinn, Jayden Daniels, Zach Ertz, Bobby Wagner, Frankie Luvu, et al all said something akin to, "It sucks, it just sucks (Daniels asked for a pardon for using the word, "sucks."), it hurts. We will not have the same locker room next year, but those who are here need to learn from it, take it into next season, and move on."
Be all that as it may, After 25 years of suck for the Washington franchise (whatever their moniker), 14-6 and a NFCCG appearance, is a whole lot better than bouncing between 3-13 and 10-6 division winner (but mostly on the 3-13 end), with little hope of advancing. I can see why you, as a Giants fan, would want the feel good story in DC to be over, but I don't think it is. Considering the results this year, the draft, their QB, coaches and front office and cap room, this appears to be the opening chapters.