Deathwing
<Bronze Donator>
Board clears, now or even with your proposed change, need hard tech counters. I hate playing control decks where you play on-curve minions and they just keep clearing them, mostly through board clears. It's one-sided uninteractive bullshit play and if you want more tech against aggro, I want clearly defined tech against board clears. And no, that's not just vomitting out more minions, I agree that is just as demoralizing to the control player.5 mana seems to be the slot where aggro decks avoid taking the card (belcher/healbot). Cards that are anti-aggro and cheaper would just find themselves in aggro decks, too.
Aggro is never going away. That's a good thing. I would surely hate for every game to be decided on who nags their C'Thun first. That meta is probably more frustrating... playing a great game just to get out RNG'd in the end.
I think biggest issue is board clears don't match the ability to refill the board. Clears are way too expensive. It's been a constant issue in this game. If board clears were cheaper than the deathrattle/sticky meta would never have been so bad. Right now it's a problem drawing clears in time, then gaining initiative after the clear. With expensive clears like Flamestrike, Blizzard, Twisting Nether, or even Nova, Brawl, ExEvil you rarely have enough extra mana to play a decent minion (6+ mana).
Doubt Blizz ever changes their mindset on it, but imo that's biggest problem. My decklist above has 6 AoE clears/dmg... but they are all rather expensive, except doomsayer. Even with so many clears it's not always enough or it's just too late.
BGH is an excellent example of a tech choice if you were seeing lots of control decks, but it's for big minions only. Give me something that punishes board clears and I think balance has a chance.