Is DAI the only DA where you can play as an elf?

No. In the Origin you can try to play as human mage/noble, elf mage/city elf (from alienage)/dalish, dwarf noble/casteless. In the DA2 yes, you can play only as human, only as Hawke.
Let's try, worth it! 

DA2 was the only game where you HAD to play as a human.

Well, there's very strong Origins nostalgia. A century or two from now and robots have enslaved the human race, there will be fans pleading with their cybernetic overlords to bring back the Warden.

Robots: Do you also want us to stop whipping you?
Fans: Well, yes. But the Warden thing is more important!

I see your experience of the Witcher forums matches mine - and I didn't even touch such a hot topic.

Having said that, I prefer to play as a human because it appears to be most honest to me. Elves and Dwarves - even qunari - are actually humans that look a little different and come from different cultures. They're all human in everything that counts, so why should I have a preference here except an aesthetic one, or one of background culture?

Now if I could play a *really* non-human character - within DA, a spirit comes to mind, if written competently - I'd jump at the chance, but between standard humans, short stocky humans, humans with pointy ears and big humans with horns I choose the standard human.  In DAO, I played a City Elf among others, and that *was* very enjoyable, but again, it was because of the background culture and the specific plot of the CE origin. Safira Tabris never felt non-human to me.  

+1. There was another thread not too long ago about how the city elf origin was the most underrated of the bunch. Couldn't agree more.

Come to think of it, the only disappointment I had with the DAI character selection was that the elf wasn't a city elf.