Yui only appeared during a sidestory in the second light novel so we can say that she didn't play a role in their decision.
Basically Heathcliff knew that Kirito and Asuna felt a responsibility for clearing the game seeing how they were some of the strongest players.
Asuna then suddenly removed her smile and muttered:
“Up until now, I had only fought on ...
According to the light novels they are cousins.
Kirigaya Minetaka and Midori, husband and wife’s own child was
Suguha, while Kazuto was Midori’s sister’s, this meant that he was the
child of Suguha’s aunt. Her aunt and her husband, without a choice, left
their only child, who was not even a year old, and passed away in an
accident. The serious ...