My take on this is that I've always thought that trying to complete the Quest should in some way restrict playing. But battlecry is so common keyword and some battlecries are so powerful that playing against quest Shaman that is trying to complete their quest doesn't feel any easier at all. Nerfs suggestions would be either change cost of some lackey generators (Sludge Slurper and EVIL Totem) or make the quest goal to be 10 battlecries. Anything less wouldn't be enough.
It does restrict the deckbuilding though. You're forced to have a deck with all battlecries (with the exception of the mogu/mutate package), which basically means you include every good battlecry in the game with very little flex space.
The quest isn't even the biggest issue here. I can't even count the number of games I won with an early mogu/evolve. I think a suitable nerf would be minions like mogu keep their cost for evolve purposes at the cost it actually took to play them. So if you play for 2 mana then evolve it, it becomes a 3 drop. There's still pretty broken evolve combos like evolving giggling inventor, but that happens at turn 7 or later. Someone highrolling a tirion on T3 is ridiculous.
There is still rabble bouncer at 7 mana reduced by enemy minnions, and some other cards that can be cheated out. It is not necessarily evolves fault that minnions can be cheated out earlier.
Problem in my mind is that by turn 5 most of the time you have quest completed while generating cards.
Totem,rat into random lackey which might be lackeys that gives spells, 2/3draw if you have quest active etc. Your hand size does not reduce too much to forcefully complete quest.
Now if you want to complete old deathrattle quest your hand is ussualy depleted fast, if you wish to complete mage quest ASAP your hand size is reduced fast as well. Druid loses board and has to come back onto board of he aims to complete quest by turn 4 with coin.
Shaman on the other hand, generates cards too easily, which leaves him not losing tempo or much value from card generation. And those same cards increase in value post quest completion.
Does it need a nerf?
Before priest gets changes/nerfed for divine spirit inner fire combo probably no. When the nerfs happen yes shaman needs massive changes at the same time, in order not to become rampant t0 deck.
For now just wait and see.