The goal is to draw your whole deck, empty your hand, play Mecha'thun and then kill it with Keen Reflex. The quest helps reduce the cost of your cards.

There are two copies of Keen Reflex, in case you overdraw one or have to clear your own board to be able to empty your hand.

Consume Magic cannot be played if the opponent doesn't have minions, so make sure to play it well before the end game. As it's quite useful in removing buffs and nasty deathrattles, I think it's worth having this card in the deck even with the risk of it getting stuck in your hand.

Malygos the Spellweaver often completes a quest stage by itself. If you complete the first or second stage this way, you get discounts on all of the draw cards, even if it's more than 4/5. If you complete the third stage with Malygos, you may lose the quest reward if you don't have hand space for it. Depending on how many cards there are left in the deck, that may or may not be a problem.

Mulligan

Keep the quest and card draw cards.

Replacements

Mecha'thun is essential to this deck list, but I assume there are other ways to OTK the opponent after drawing your whole deck and discounting significant parts of it. The advantage of Mecha'thun is that the final combo is just two cards, which makes hand management easier and is less vulnerable to disruption.

Final Showdown isn't strictly required, but the discounts do make the game plan much easier.

The other collectible epic and legendary cards are all optional:

The Curator is a tutor for Coldlight Oracle and Malygos the Spellweaver (although Alexstrasza the Life-Binder can also be drawn), to keep your card draw going.

Lorekeeper Polkelt will shuffle the deck according to the cards' original costs, so it puts Mecha'thun and the dragons and demons on top of your deck. This is useful if you have drawn a lot of cards already and just need a bit of survivability.

Pit Commander is useful to thin your deck and tutor the armor of Mo'arg Forgefiend.

Artificer Xy'mox gives you the ability to draw two cards for one mana, or to not draw any card if your deck is already depleted but you still need to empty your hand.

Altruis the Outcast is a cheap board clear for opposing minions. Opponents also tend to prioritize removing it, which gives you a bit more time to execute your game plan.