See p. 2 of this article. The main point in understanding what's happening is not specifically related to Tachanun, per se, but, what's so important about Monday and Thursday that caused an increase in Tachanun as well as supplications.
Summarizing what the article states, Levush Techelet commentary Orach Chaim 135 explains the ruling established by Ezra on why Torah should be read on Mondays and Thursdays. Levush states that anyone who goes moe than 3 days without reading Torah has enemies attacking him. Therefore, not only were these days established as days to read Torah, but also, these are days when you increase supplications which includes among them, Tachanun, Av Harachamim before reading the Torah, and the Yehi Ratzon after reading the Torah. The point being, that it has nothing directly to do with Tachanun, but is a=included in the notion of increasing requests on supplications on Mondays and Thursdays.
I would posit, therefore, that on Yom Tov, Rosh Hodesh, and other celebratory days, when there is a Torah reading, that Torah reading would occur regardless of which day of the week it is. No Tachanun is said on those days, anyway, and we are also pre-empting the normal parsha reading that's usually done on Monday's and Thursdays. Also, the celebratory nature of the day, would override the necessity or appropriateness of these extra supplications.