Is the reason for relying on leniencies for agunot is because the lady wants to remarry and it's not good for her to be single? (so in a case that the lady is not desperate and could manage on her own, is there still a Mitzvah to try to fetch up any type of leniency to permit her from her aguna status?)

We are taught from the beginning of the very first man, “it’s not good for a person to be alone!” (Breishit 2, 18). Everyone needs a soulmate, and the special desire of women for companionship, we believe is essential and inherent (Kiddushin 7a). In addition, the Jewish family life is central to Torah living. Accordingly, even if theoretically a woman were to say that she doesn’t want to remarry, we would do her a favor and enable her to do so. Nevertheless, technically, if she doesn’t want to marry, and doesn’t even approach a Beit Din on the issue, obviously the Beit Din won’t deal with her status “against her will”, and she can remain an aguna if she so desires.