I think, based on the specific wording involved, that a self-protecting EMT will still die to the Outrider. Let’s compare the wording on said Outrider to a killing role, AU Batman.
Outrider
“Each Day you may target any player of your choosing. If that player is under the protection of any active protecting role(s) - the EMT or Doctor Seward, for example - the player(s) doing the protecting will die. Your target will still survive any attempt on his life unless the protection is obviated in some other way.”
AU Batman
“You are a vigilante. Each night you kill someone. That’s it. Pick someone you think is anti town and kill them. You have the power and the training, oh and you have the drive too. You don’t get to not kill someone, and you aren’t suicidal so you can’t kill yourself. Make Daddy proud.”
Do you see the difference? AU Batman kills, but the Outrider doesn’t, their target just dies. That would say to me that being protected from kills doesn’t stop you from dying.
Of course, that could just be me applying Magic the Gathering wording rules to something unrelated. But it’s logical at least!