There have been some infamous combo decks in Hearthstone's history. The most well known is likely the old Midrange Druid that utilized Force of Nature and Savage Roar, but it's not alone as a deck that seeks to combo you down quickly. Others such as Freeze Mage and Miracle Rogue seem to be acceptable in the eyes of Blizzard as players have more ways of interacting with them.
Hearthstone is fun because each game is a little different from the last. Combo decks make for very different types of games, where players can’t rely on their normal decision-making heuristics and have to reconsider their strategic approach to the game. So, in general, combo decks are good for Hearthstone because they add some texture to the ladder experience. But like any other deck, combo decks that become too popular cause issues.
When we make cards like Emperor Thaurissan and Counterfeit Coin, we’re aware that they tend to enable combo decks. We don’t usually set out to make a particular combo deck be a particular power level; we are always worried about missing and making a deck more powerful than we’d like – and that goes for any type of deck, not just combo decks - but that doesn’t mean that we don’t want any combo decks at all. We do want combo decks. We just want them at the same level that we want other decks.
Specific combo decks can be problems for the same reason that any other deck can be a problem: when a deck in Hearthstone gets too popular, you play against it so frequently that it stops being fun. Further, usually popular decks are powerful, so you are also likely to be losing more games than you win while also playing against the popular deck more often than you would like.
Powerful combo decks tend to exacerbate this problem because most combo decks aren’t trying to interact with their opponents on any axis that involves minions. It is cool when, once in a while, you play a game against a deck that is all card drawing and removal and you have to consider how you want to ration your threats instead of finding little victories in profitable trades. It is less cool when you start playing a substantial fraction of your games against “Frost Nova, Doomsayer, go” before getting Ice Lanced out of the game. Figuring out a good trade is more fun than making educated guesses about how your opponent will kill your minions over the next few turns. I am not saying that some or all combo decks are inherently bad for Hearthstone.
I am saying that when a deck is popular, it becomes less fun to play against. Popular combo decks have the further problem that they try to ignore opposing minions as much as possible, which is frustrating for most non-combo players.
This is true whether or not a deck is easy or hard to play, although the decks that are hard to play tend to be less popular because less skilled players don’t play them as often.
Also, saying that a combo deck is interactive because it has a bunch of removal is true only in the loosest sense of the word. Combo decks use their removal to try to reduce their interaction with their opponents as much as possible.
Cards are interactive when they generate strategic options for both players. Minions are interactive because their controller has options on how to leverage their threat and their opponent has options on how to remove it. Removal itself reduces the strategic options for both players: it reduces the amount of stuff in play that can be interacted with.
This isn’t to say that all removal is problematic – removal spells are very important for Hearthstone – but I see the idea of ‘this deck is interactive because it is really good at killing minions’ frequently and I wanted to challenge that assertion.
Most combo decks, in addition to trying to avoid interacting with minions, also try to avoid letting their opponent interact with them. The problem with OTKs isn’t so much ‘I was at 30, then I lost’ as much as it is ‘I was at 30, then I lost, and there wasn’t anything I could do about it.’ You can play a taunt minion against a Leeroy combo, but again, most combo decks are very good at killing minions, so the idea that a taunt minion will save you against a Miracle Rogue that’s drawn their whole deck is a stretch.
So, most combo decks try to avoid interacting with their opponents as much as possible, and then win in a way that is extremely difficult for their opponent to interact with in a meaningful way. It’s good when those types of decks pop up on ladder occasionally. But when those types of strategies are too popular and powerful, they are frustrating, and we nerf them.
When they are not popular, we tend to let them be. Usually they are less popular because they are less powerful; often they are less powerful because they have to interact with their opponent. For example, the Aviana-C’Thun combo has to play a lot of C’Thun minions, which facilitates interaction for both players. Combo decks that aren’t super popular or super powerful are great to have around; as I noted earlier, they do a lot to add variety to the ladder.