Why do we Daven Yom tov Musaf on Shabbat Chol Hamoed and do Yom Tov Leining? Based on תדיר ושאינו תדיר I would have thought we say regular Shabbat Mussaf with insertions for Yom Tov and Similarly continue with Parashat Hashavua just with the Yom Tov Maftir.
You mention that the reason you suspect we should use the Shabbat text of the musaf service is because of the halakhic principle called
תדיר ושאינו תדיר, תדיר קודם. This principle means that when you have two mitzvot to do, one of which is done frequently and the other of which is done infrequently, all else being equal one should do the frequent mitzvah first.
An example of this is brought in the Kitzur Shulchan Arukh 76:12 which says that if you delay in davening musaf until the time comes for mincha so you're left with needing to say both tefilot, you should say mincha first because we daven mincha every day and tadir v'she'eino tadir, tadir kodem.
This halakhic principle wouldn't seem to apply in this case because we only say the tefilat musaf once so there's no concept of "first" or "second" (although every time the day is mentioned, shabbat actually always does come before the chag so it seems the principle is adhered to there). The principle of tadir has nothing to do with which text should be used.
Musaph of Yom Tov is used throughout Yom Tov, whether it is Hol Hamo'ed or not. It has nothing to do with it being Shabbat. The reason seems simple. The Musaph prayer replaces the actual Musaph sacrifice. On any holiday, this Musaph sacrifice was offered throughout all the days of the holiday.
Of course, on Shabbat, we include the mentioning of the Shabbat Musaph offering along with the holiday one.
If your wondering why not use the Shabbat Musaph text and just include the paragraph for the Yom Tov Muspah offering (i.e. the one paragraph) - I would surmise that an integra; part of the holiday Musaph text includes mentioning of aliyah laregel - the mention of the "pilgrimage" - which is absent from the text of the Shabbat Musaph.
I'll have to research something that addresses the Torah reading.
The musaf we pray include shabbat and chag hamatsot and shabbat (or hamanoach following Rashi and the Shulchan Aruch) is mentioned before chag hamatsot in each occurence, without any contradiction with the principle of tadir kodem.
See SA OC 490, 9:
ובמוסף אומר סדר מוסף יום חול המועד, אלא שמזכיר של שבת, ואומר: "ותתן לנו את יום המנוח הזה ויום חג המצות הזה", וכן אומר: "את מוספי יום המנוח הזה ויום חג המצות הזה", וחותם "מקדש השבת וישראל והזמנים". ומפטירין "היתה עלי":
... he say this day of musefey (plural of musaf) ...
See Gemara Betsa 17b with the comment of Rashi:
ובמוספין מתחיל כו' אלא בהא הלכתא כרבי דאמר אף חותם בה מקדש השבת ישראל והזמנים ...
On musaf he begins (with the Sabbath [formula] and closes with the Sabbath [formula], and makes mention of the holiness of the day in the middle... Said Rav Huna: The halachah is not as that pair [of scholars]) but the Halacha is as Rabbi said, that he concludes mekaddesh Hashabbat, Israel Vehazzemanim
In conclusion, the Musaf is a double musaf of Shabbat and Yom Tov and Shabbat is mentioned before Yom Tov in each occurence that both are mentioned.