Except that they were all wrong on that turn also since if they Alex'd face the opponent they would have won:
1.) TL would be at 15+2, then gain 3hp from hero power and lose 2 from fatigue.
2.) They can't do enough damage to the Alex to kill it, but they must kill this to avoid getting killed later. So they hit face for 4 dmg and shoot alex for 3 dmg, healing 3.
3.) GC now at 8hp with 1 card left and TL with 18hp. Alex hits face for 8, fatigure for 3, ping for 1. GC at 8hp, TL at 6hp. Fireball would be lethal +6, arcanist could be played if drawn.
4.) TL can now kill GC's Alex or they could leave GC at 1hp with 0 cards left. In either scenario they die next turn, to kill Alex the hero power must be used on it. Leaving them at 9hp, next turn they take 4 dmg from fatigue meaning fireball is lethal again even if it's the last card.