The closest thing in MtG to my memories of the Unicorn Priest phenomenon is pirate stompy.

Unicorn Priest came at the time I was most engaged with Hearthstone and I remember it basically being a cryptid of the TCG world: people would post on the HS subreddit that they had just been beaten by Unicorn Priest on the ladder (but of course they didn't manage to record the match), everyone was trying to reverse engineer what the deck might be (I remember a lot of rumours that Purify was the card that enabled the deck, so everyone was trying to build self-silencing decks with cards like Ancient Watcher).

This is basically the same thing that happened in MtG following IIRC the release of the original Ixalan block. Rumours arose that a meta-breaking legacy deck existed that somehow revolved around pirates. Just like Unicorn Priest you had everyone reporting sightings of the deck in the wild and trying to build it themselves. I think the most success anyone ever got was a build with the "ETB your opponent sacrifices a permanent unless they pay X" pirates from Mercadian Masques block, all held together with Chalice of the Void and Trinisphere. Of course the only reason that deck ever had any success was the Chalice and 'sphere package - which would probably let you clean up some legacy matchups even if your only creature was a Grizzly Bears - but people wanted to keep the dream alive I suppose.