Seemingly, davening on special days should reflect our primary experience of the day. As such, on Shabbos we say “mekadesh haShabbos”. Similarly, on Rosh Chodesh we say “mekadesh Yisrael veRashei Chadashim”. On Rosh HaShana we both name God as king of the world and name the day (Yom HaZikaron, its biblical name) and on Yom Kippur we not only name the day and proclaim God king of the world but also celebrate his forgiveness. As such, one might have expected that for the Shalosh Regalim (Pesach, Shavuos and Sukkos) we would name the day at the end of the bracha in davening, the most important part of the bracha. However, we don’t. Instead, we have a generic label for all three holidays which seems to proclaim their sameness, not their uniqueness. Furthermore, the entirety of the davening for these three festivals seems the same. While we do (twice) name the specific festival we are celebrating, the wording of the prayers doesn’t distinguish between them at all. Why is this? It would seem that the primary of experience of Yom Tov (which our davening should reflect) is just that, a Yom Tov.
An interesting test of this premise could be the Gemara in Eiruvin 96a. There the Gemara states the (normative) opinion of Rabbi Akiva that one is exempt from the Mitzva of Tefillin on Shabbos and Yom Tov because Tfillin is an os, or sign (according to the Rid it is a sign that we keep the mitzvos) and so too Shabbos and Yom Tov are an os (from God to us). The Tosafos there ask whether this applies to Chol HaMoed because it features a very limited probation on melacha as well as the mitzvos which characterize the festival such as eating matzah or sitting in the Sukkah. They prove from a Gemara that seemingly there is no such exemption on Chol HaMoed. Seemingly, if these mitzvos were sufficiently a sign so as to preclude the need for the sign of Tefillin they would be more prominent in our davening as well. As such, it would seem that they form only a secondary layer to our experience of Yom Tov.
It should be noted that the Behag and the Zohar both state that Tefillin should not be worn on Chol HaMoed seemingly contradicting my analysis. Also I think that what I wrote here explains nicely the reason why Chol HaMoed our only mention of the chag features its particular description, either Chag HaMatzos or Chag haSukkos. On Chol HaMoed our primary experience of the festival is really through its own unique mitzvos and not through any issur melacha, and as such, that is what our davening reflects. Just some musings from Yom Tov