I tried the Hearthstone Monster Hunt and completed it with one of the four classes. I may someday bother to complete it, but it is not exactly good or recommended.
Compared to Dungeon Run, there seem to be more occasions for smaller choices in your cards, rather than a few huge, random swings. The Tracker’s hero ability is to pick one of three spells already played. The Cannoneer makes minion placement matter more than usual. The Time-Tinker lets you choose when to reset randomization. There are more effects recycling and Discovering cards, more smaller cards, fewer instant win/lose conditions in your deck.
Compared to Dungeon Run, the boss decks have even more huge, random swings, making all that irrelevant. You will faceroll some, you will get stomped by others, and occasionally all those smaller choices will make the difference. But mostly, it’s Hearthstone, it’s random. Some games would have run from a reputation like that, but Hearthstone runs with it.