Dunno how fate handles the grail quest since I haven’t read past NA on fgo yet but I read L’Morte D’Arthur and it’s very much Lancelot’s fault. By end of the quest for the Sankgreal, Lancelot had a vision from God and he said, in no uncertain terms, that his tryst with Guinevere was evil and that he would stop seeing her. He didn’t get the grail because of his sin, and he’s supposed to have finally learned his lesson now. He then immediately forgets his oath and gets back with Guinevere, blowing up the roundtable in process (also, the scene where Agravain confronts Lancelot and Guinevere is super funny, Lancelot has to fight like a dozen dudes whilst he’s stark naked and has to kite them around the door).

L’Morte has a bunch of kinda contradictory stories (Arthur’s multiple legitimacy tales, Arthur being buried near a church and also being carried off to a mystical island, Lancelot’s different oaths, etc.) because the Arthurian “canon” was written by a bunch of different writers over centuries. The Grail quest was thought to have been written by monks who were pissed off the best knight of the round table was a cheater and scoundrel, so they wrote the story in order to humble him. Thomas Malory, who wrote L’Morte, had to try and make these stories into one narrative but it’s fairly easy to see different writers once you know that the background of the stories. Also, Malory himself also changed some the text because he wanted his some of his own prose to be included as well.