About Lan apologizing to Moiraine after Everything:
"I'm sorry it took me so long to do my duty."
I don't think this is Lan - or the show - saying that Lan was the one in the wrong, after how Moiraine treated him so badly.
I think it's very clear from the show (and as other ppl's metas have pointed out) that Lan and Moiraine both know EXACTLY what she is doing: she's trying to push Lan away because she is in pain
(Ofc, Lan might well not know that this is also her trying to keep him safe, but he also might.)
Instead, I think it was him…making a point of reclaiming his role as her warder, so to speak. It was him saying "I know you tried to break what is between us, but I will not let you."
It was him saying, "I wish I had figured out how to help you sooner."
It was him saying, "The time for conflict between us is past, because the Dragon Reborn is here, and it is time for both of us to do our duty."
It was him saying, "I am your warder, and I am done failing to protect you."
It was him saying, "I am in this fight alongside you, and you are not going to leave me behind again."
It was him saying, "I don't care what happened to us, these past months. My place is besides yours."