It's how you have to manage the situation. The Nationals could only win that game in two ways (really one but the difference matters) - scoring >3 runs in an inning or scoring 1-3 runs in an inning. The scenario where they score 1-3 runs is much, much more likely so that is going to require a save. If you pitch Doolittle in the 9th then you have to put either Romero or Perez in the even higher leverage situation that a save situation presents.
I'd like to see leverage charts that say a situation where the next run = loss, as in the tie in the 9th on the road, is higher leverage than a "save" situation in extra innings. I don't believe it, but I could be wrong. Sure, it may be more deflating to let a game get closer, tied or lost when you have grabbed a lead in extras, but in terms of runs damaging your hope of victory, giving up a run when that loses the game is a definite killer, while giving up 1 when you have a 1 run lead just means more overtime.
Of course, Dusty agrees more with you than me, Janes says:
So Baker, who was going to use Sean Doolittle in the ninth if the Nationals had a lead, but Romero if they didn’t, chose Romero. When the lefty tried to elevate a fastball to Pollock, it stayed down the middle, then flew out to left center. A few intentional walks and one out later, Drury singled through the right side to end the game
https://www.washingtonpost.com/sports/nationals/bullpen-falls-apart-again-for-nationals-in-6-5-loss/2017/07/22/5b2b88bc-6e9b-11e7-9c15-177740635e83_story.html
