I am piecing together a few different ideas so I apologize if I misstate any of them.
According to the gemara in Shabbat 153a, one should repent a day before his death:
(l) (Mishnah - R. Eliezer): Repent one day before your death.
(m) Question (Talmidim): One does not know when he will die!
(n) Answer (R. Eliezer): Therefore one should repent today, lest he die tomorrow - this way, he repents every day - "B'Chol Es Yihyu Vegadecha Levanim v'Shemen Al Roshcha Al Yechsar".
One of the essential components of repentance is confession (and, it seems, a sincere statement of regret). This statement of regret and a will to change appears to be part of the tachanun prayer, and the confession is what the Vidui is all about. The two prayers, said in shacharit after the amida seem strongly intertwined (as per the comments here).
However, there are days on which we are forbidden from saying tachanun (and, it would follow, Vidui).
Wouldn't that mean, then, that on those days, we cannot do full teshuva and cannot comply with the mishna in Shabbat 153a?
Why would we institute a system of repentance and indicate that it should be a constant practice, and then proscribe its essential steps on certain days? Is repentance contraindicated on happy days?