It's too bad they can't use the phonetic alphabet due to the word being Delta - those words were all chosen for good reason, they're very distinct.
I used to do phone tech support as an internship during college and would sometimes have to read out passwords to people.
Our team used to always just pick random words because there was no official policy. Eventually someone found the phonetic alphabet and we all started using that because it was so much easier.
My guess is that Alaska didn't give them any specific training on words to use with the announcement. I've seen some do the "A as in Awesome..." route (that's as far as I usually make it) but those gate agents may have just come up with that on their own, or shared between themselves. In this case, maybe he meant "depth" or he just thought "death" would be amusing but yeah, not a good call.