FWIW, I can give you an idea that I developed with a friend of mine. I don't have an actual source, though this idea is based (however loosely) on the Ramban.
In short: Ya'aleh Veyavo is a prayer asking for God to judge us favorably, which is appropriate for the holidays and Rosh Chodesh because they are all mini-judgement days.
By referring to Rosh Hashana as a day of זכרון תרועה, The Torah is telling us that, on this day, we need a זכרון before God, which we accomplish by blowing the shofar. According to the Ramban (Vayikra 23:24, and in his Drasha for Rosh Hashana), this is a reference to the judgement which occurs on Rosh Hashana, and the 'זכרון' that we are seeking refers to a favorable judgement on that day. Hence, this prayer of Ya'aleh Veyavo is using the Torah's terminology (זכרון) for asking God for a favorable judgement, for us, the Jewish people, Yerushalayim, Mashiach, etc. - a perfect prayer for Rosh Hashana.
However, Rosh Hashana isn't the only time when this prayer is said, because Rosh Hashana isn't the only time when the world judged in some way, as the Mishna/Gemara Rosh Hashana states (daf 16a), that on Pesach we're judged for produce, etc. when, during the time of the Beis Hamikdash, we would have performed mitzvos to accomplish something similar to the זכרון as the shofar accomplishes on Rosh Hashana (Ramban in Drasha Rosh Hashana). Thus, the holidays are also a time appropriate for such a prayer.
Rosh Chodesh as well is a mini day of judgement. There are several sources which indicate this, but just to quote one: the Levush (O.C. 422:3) writes that the reason why we don't say a full Hallel on Rosh Chodesh is because it is a day of forgiveness, and therefore הוי כמו ראש השנה
ויוה"כ שהם ימי דין, "it's like Rosh Hashana and Yom Kippur which are days of judgement".
While it's true that there are other aspects of each holiday besides for the judgement that occurs on those days (such as the mitzvah of simcha), those also get their mention in the holiday prayer. But even if the דין is only tangential to the message of the holiday itself, it makes sense to be something that we'd want to daven about.