Big Game Hunter is a big deal. Hearthstone is powerfully shaped by its very existence, and in fact there are plenty of cards whose lack of inclusion in decks can be traced to BGH's influence. That doesn't mean that BGH is in every deck of course, but it is in enough decks that many legendaries that have 7-8 Attack would actually be better off with 6 Attack because then they would not be a BGH target.
There is a delicate balance in Hearthstone where just enough 7+ Attack minions are played to justify some BGH inclusion, but should the metagame ever evolve to contain many more 7+ Attack minions BGH usage would jump up and then those big minions would suddenly become terrible again. Letting your opponent blow up your 8 mana card for just 3 mana and then get a 4/2 body on top of it is often too much of a tempo swing to survive. The only thing really keeping big dudes in the game is Dr. Boom because he is so overpowered that even with BGH out there Dr. Boom is worth playing. There is basically a BGH equilibrium, whether we like it or not.
I think this is a sorry state of affairs. It is definitely good that people have to consider how many expensive spells to play, as you don't want every deck to just load up on 8 cost monsters, but there are plenty of aggressive decks that will punish anyone so foolish as to attempt this. The nice thing about aggressive deck is that punishment is very granular and you can carefully work on your deck's mana curve to make sure you aren't getting too greedy. BGH, on the other hand, basically just guarantees that it is specifically the 7 Attack minions that are bad. 9 cost dragons that are 4/12 are just fine because although they are slow you can't get blown out by BGH like you can with an 8/8. BGH really warps the high cost minion market in a way that I feel is bad for the game. Nothing else in the game suggests that having a huge Attack number is bad, but BGH really enforces that.
Given that BGH is in the Classic set he is destined to be around in Hearthstone forever. It is fine to have tech cards in the basic set, in fact it is ideal, but I question whether a tech card that is this powerful is a good idea. If for example people really only used BGH to fight against Handlock decks then it would be totally fine because it would be a rare choice used by a few decks that have rough Handlock matchups. Unfortunately it is used against everyone, and that is not a restriction I would like to see in the game forevermore.
So what can we do about it? I wouldn't want BGH to be nerfed to uselessness, so one simple option is to make it a better baseline minion and reduce its Battlecry in efficacy. For example, if it did 4 damage instead of outright destroying the minion but had its base stats upgraded to 4/3 I think it would be a great card but much less swingy. Far better to play on turn 3 if you have to, and far less brutal on turn 9 to kill a giant monster.
Unfortunately that solution still leaves BGH in a spot where it makes 7+ Attack minions much worse than those with 6 Attack. I don't much like that breakpoint, so I came up with a few options that get away from it. The first is mathematically pleasing but clunky to implement: BGH does 1 damage to a minion for each Attack it has over 4. So a 5 Attack minion takes 1, 6 Attack takes 2, etc. This is similar to the previous example but isn't simple to explain on a card, so it probably isn't going to fly.
The other option I considered is to have BGH reduce an enemy minion's health to 4. This is pretty worthless in the early game, but is really good at doing a number on big health minions in the late game. There is no breakpoint where big minions become bad though which I like a lot. It retains the general flavour of hunting big monsters, reduces BGH's overall swinginess, and gets rid of the 7+ breakpoint that irritates me so.
I don't know that BGH is on Blizzard's radar, but I really hope it is. There are a lot of really fun 7+ Attack Legendaries out there and I want them to see the light of day.