I've been going back and forth on writing this thread for a couple of months because I couldn't find a way to frame it that I was happy with, and if I couldn't convince myself I'd never convince anyone else. But the early Stormwind meta has added some clarity to the picture, so here goes...
I'm pretty sure the state of card draw is one of, if not the biggest problem in Hearthstone right now. Put simply, its more efficient than is good for the game. There's too much of it, it's too cheap, and it too often tutors key cards. It's not just the classes that are supposed to be great at drawing cards, its everyone. To varying degrees sure - rogue is still better at it than most - but not to much significance. Even the classes that are supposedly bad at it have card draw that would have been powerful enough to be confined to classes like rogue a couple of years ago. Not to mention super powerful neutral options that guarantee any class has draws for days.
Take Elven Minstrel for example. It was a good card in a very strong expansion in a class that has always had good card draw. Compare it to the likes of Refreshing Spring Water and Primal Dungeoneer, both of which are in classes that were traditionally so-so at drawing cards, and the Minstrel is laughably weak.
Why it's an issue
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OK, so there's a lot more card draw, why is that a problem? At it's heart, the problem is that card draw benefits different archetypes to wildly different extents. In particular, aggro can avoid running out of steam (as demonstrated in the Barrens meta, and quite a lot of last year too) and combo can find everything really quickly (as demonstrated in Stormwind). Normally that wouldn't be too much of a problem because card draw is slow so aggro can't afford to use it and combo has to be careful not to fall too far behind when using it, but it's all so cheap these days that it's not a problem for either archetype.
Meanwhile, control and value benefit relatively little. Both try to get as much as possible out of each card in the deck and clog up their hands if they draw too much. Plus, their win conditions barely exist when they cannot exhaust aggro's resources and they die too quickly to put any pressure on combo.
Control can at least try to respond to heavy card draw in aggro by running more removal and making use of their own card draw as best they can. That didn't work super well in the Barrens, though priest at least managed it. (It was a horrible deck to play against, but that's besides the point.) I was actually hopeful the defensive tools in Stormwind would make up the difference though. Perhaps they did? We can't say because Stormwind made a bunch of powerful combo decks that, thanks to the copious efficient card draw, can completely mess with the natural order of things and win as fast as aggro does.
In an ideal world we have the cycle: aggro > combo > control > aggro. In Standard things are set up so aggro = combo >> control <? aggro. Wild has been like that for a long time, and it's an inevitable result of lots of great cards existing, but Standard should be small enough to let control function.
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A more subtle issue is that it frankly makes the game less interesting. Decks always find the right cards, reducing variety of gameplay. There's no hope of the opponent not finding their key cards, so your decks have to be able to handle a high-roll every game. That's utterly crippling for meme decks.
Plus, from a design standpoint, classes like rogue now often receive card draw as their main payoff for synergy because that's how they establish themselves as the classes that are great at drawing in a world where everyone is. Fine, but I want to see fireworks and interesting effects, not the hundredth bloody way to draw more cards when I already have 5 times more of that than I know what to do with.
How did we get here
It's difficult to say for sure since I'm not part of the development team, but I can identify a few possible causes, all of which occurred at a similar time.
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First, we had an overload of card generation in the Year of the Dragon. By the time the devs decided they had taken it too far, they had already dug the hole: how do you introduce decks that can compete with that level of resource generation without doing the same thing? Well, give them lots of card draw instead. Unless the game goes to fatigue, it will serve the same purpose.
That's all fine and dandy during the Year of the Phoenix when excessive card draw and generation existed side by side, but once the Year of the Gryphon rolled around we had some classes that weren't given the strong card draw during Phoenix and no longer had the value generation to compete with. This is where the problem really became apparent to me: shaman needed more card draw to compete with everyone else, but the problem wasn't really that shaman had too little, rather that other classes had too much. But by this point it was too late. It would take sweeping nerfs to Phoenix's draw to resolve the issue properly, verses adding 1 super strong bit of card draw to shaman. The latter route is the easy quick fix, but committing to it fundamentally changes the balance of archetypes.
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There was a related issue that is specific to aggro: demon hunter was introduced as a figurehead for the change in philosophy from value to draw. It immediately set a high bar for how much card draw a class that's good at drawing has, and continued adding to it as more expansions came out. Other classes with great draw followed suit, with rogue quickly finding itself with great card draw for every conceivable occasion.
So what, 2 classes that are meant to be good at card draw have good card draw, surely that's sensible? Well, yes, except that those 2 classes are also high tempo, aggressive classes with hero powers that help win the board. So other aggro decks struggled because they would lose on both tempo and resources. They can only really catch up if their own draw is improved substantially. That was achieved via both neutral and class cards. In any case, the end result is that all aggro decks now need to have good draw because otherwise rogue and DH will stomp on them.
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There was another factor at the control end too, especially with priest whose card draw was gutted in its Classic set rework. In line with its class identity of being poor at card draw, it was stripped of its primary sources: Northshire Cleric and Power Word: Shield. This didn't work out too well for the class, which performed fine overall but only through obscene value generation that the community complained about (it is no coincidence priest had the last surviving bulls*** value deck).
The reaction from the devs seems to have been that perhaps no class should be bad at card draw after all, which freed them up to print good card draw everywhere. I think the mistake here was not class identity, but how they handled priest's Classic rework. Classic always had cards outside of class identity to ensure every class had at least some options to work with. You might be able to strip away some things for the sake of class identity (e.g. they could take away rogue's AoE, as indeed they did in the Core set), but card draw is too fundamental a mechanic to take away. Leaving priest with only neutral options that don't compliment the class was bound to hit it harder than intended.
(For the record, I think priest's card draw is actually in the most sensible spot of all classes right now, so they've been doing something right.)
What's the fix
If I'm optimistic, a few nerfs to the classes quickly completing game-winning questlines will open up enough space that control can at least test how effective the defensive tools in Stormwind are against aggro. That might be able to reset the archetype match-ups.
Failing that, the best solution probably requires a lot of patience and a change in philosophy from the developers. I say change, but it's really just taking a few steps back. Just as they decided they went too far with the amount of value generation, they would need to believe they have gone too far with card draw too. Tone down card draw in the best classes (DH, rogue and warlock), and let other classes be kinda bad at it again. I appreciate that is quite difficult when DH has a questline that requires crazy card draw, but still.
They cannot realistically nerf their way out of this problem since it affects way too many cards, so we'd probably need to wait for things to rotate.
What I fear is that they 'solve' it through more power creep, by giving control completely broken tools to combat the situation. That would be the point where I might finally consider leaving since so much of my enjoyment in the game is derived from making fun but weak cards work, which is already tough enough as it is.