When you're talking about death and the afterlife, chances are you'd do so with a rather omniscient voice (that all-seeing perspective). It's kind of unavoidable since none of us has ever been there, done that. Dylan Thomas chose this sort of speaker for "And death shall have no dominion" and it fits pretty well with all the biblical allusions, too.
With words like "shall" and "dominion," an omniscient perspective is likewise pretty ideal since Thomas's diction tends to get a bit Romantic and old-fashioned at times. Imagine if we had a very limited perspective through the voice of a teenager in the 21st century. The waves crashing on the seashores and that flower lifting its "head to the blows of the rain" wouldn't sound quite right.
So having an all knowing and seeing speaker makes the poem jive with its deathly, cosmic content while also serving to prove Thomas's point that "death shall have no dominion." We kind of need that god-like voice to convince us that death isn't the end. Not to mention the refrain that sounds rather omnipotent in the first and last lines of each stanza. It's everywhere…
And since the speaker sweeps the globe in a short amount of time, that omniscient voice is even more fitting. After all, we'd be hard pressed to find a first-person perspective that could walk us through the cosmos, the sea, and some medieval torture "racks" in only three stanzas.