Considering they only accept elves joining them if said elf converts to their beliefs and culture, they pretty much do.
The Dalish are a nomadic group focused on following the elven religion and elven culture, like the Sephardic Jews.
And let's not forget the "Soon we will reeducate all the elves who have lost their way" sentiments.
You're conflating the view of some with the view of all of them, and it's also true that some Alienage elves see themselves in a similar fashion, as they view those who try to live outside the walls of the Alienage as people who throw away "being elven". As hahren Sarethia wrote, "Here, we're among family. We look out for each other. Here, we do what we can to remember the old ways. The flat-ears who have gone out there, they're stuck. They'll never be human, and they've gone and thrown away being elven, too. So where does that leave them? Nowhere."