Big Game Hunter, or “BGH” has been a topic of discussion in the Hearthstone community for a long time. Having pretty reasonable stats for 3 mana, there is little risk in including the card in your decks. The potential upside, on the other hand, is massive. Not only do you get a two-for-one exchange in terms of card advantage, but you’re spending 3 mana to destroy a 7+ mana minion of the opponent’s while developing your board. It is such a massive swing that including any 7+ attack minions in your deck at all has a lot of risk involved.
Ubiquitous cards such as Dr. Boom would actually be even better if they had their attack stat reduced to 6, simply to avoid BGH. This pushes a lot of potentially fun and exciting cards such as Deathwing, Arch-Thief Rafaam, or Inner Fire out of the meta. BGH also provides classes like Druid, who would otherwise be weak to large minions because of their lack of hard removal, with an effect they should otherwise not have. When every class has access to Big Game Hunter, cards like Polymorph or Assassinate become much more lackluster.
I would take this balance patch as an opportunity to tone down the strength of BGH significantly, so that we see more large and exciting minions being played. I recommend changing the card to only destroy Dragons, but leaving the stats and cost the same. This would also introduce a counter-measure to an archetype that does not otherwise have one.