I think the concerns people have about the new-player experience all ultimately boils down to the flawed ranked system Blizz has opted to implement:

Imagine, if you will, that Hearthstone had an MMR and ranking system akin to that of its sibling games Heroes of the Storm or Overwatch, with multiple tiers and divisions to divide players and ultimately result in a higher prevalence of 'fair' games.

This structure, combined with little to no rank decay between seasons would allow the more inexperienced players to play against players of similar skill levels and, given time, eventually move up as their game knowledge and collection grows.

Now of course a flaw in this system also exists, whereby players may intentionally lose ranks and 'farm' against players of inferior experience or deck-caliber and thereby easily grind out quests or play-time rewards (which is why a overhaul of the quest system such that it is not win-based is necessary). Granted, this problem exists today also and is part of the reason why the new player experience is so frustrating, but implementation of a protection system that detects players intentionally insta-conceding multiple games in a row and delivering some sort of punishment (make ranked locked for a while maybe?) would be well received.

Even if the current structure was maintained, the removal of rank decay (and subsequent alteration to end of season rewards) would ultimately result in a better game, for both newer and more casual players (however Legend would obviously need to be changed such that it would not be possible to have millions of ranks in legend itself, perhaps something like the Grand-Master system from OW or HotS is appropriate here, or maybe all Legend players degrade to Rank 1 between seasons).

Tl;dr. The removal of rank decay (and win streaks?) would pretty much fix everything (maybe, this would have to be experimented with).