It's important to notice that there IS a verse for Miriam in the Hoshanos that we all said (or mumbled) just a day before Tefillas Geshem, and we must remember that Tefillas Geshem and Hoshanos were written by the same author, R' Elazar Hakalir. So clearly R' Elazar Hakalir valued Miriam's role as it relates to water and mentioned it in our prayers for water, which is the main purpose of the piyutim of Hoshanah Rabbah.

This is the verse:

לְמַעַן נְבִיאָה מְחוֹלַת מַחֲנַיִם. לִכְמֵהֵי לֵב הוּשְׂמָה עֵינַיִם. לְרַגְלָהּ רָצָה עֲלוֹת וָרֶדֶת בְּאֵר מַיִם. לְטוֹבוּ אוֹהָלָיו. הוֹשַׁע נָא וְהוֹשִׁיעָה נָּא אָבִינוּ אָתָּה In the merit of the prophetess [Miriam] of the dance of the camps, to those of thirsting heart she was an inspiration, at her feet ran, rising and descending, the well of water -- for the sake of the one of goodly tents, may You save now and being salvation now, for You are our Father.

Translation from The Artscroll Siddur

So why not include her in the main pizmom of Tefillas Geshem? Perhaps some of the answers above are correct. I think there is another issue here. Tefillas Geshem (unlike Hoshanos) was written specifically for the "Avos" section of Shemoneh Esrei. It is beyond the scope of this question why the traditional tefillah mentions only the Avos and not the Imahos, but I think R' Elazar Hakalir was simply fitting into that framework, which is what he does in many of his piyutim that are written for the first brachos. Extending it to Moshe and Aharon is not as revolutionary since they are often seen as part of our group of leaders (think Ushpizin for example).

There is one notable exception that I noticed (and keep in mind that he wrote piyutim for the "Avos" section for all Shalosh Regalim and Arbah Parshios) and it is not Shemini Atzeres but rather Rosh Hashanah!

R' Elazar Hakalir felt that Rosh Hashanah was THE day of the Imahos, and chose to focus on them on this day (and given that it is the birthday of man, I think the focus on the Imahos on this day brings them up to a special place in his mind). On Rosh Hashanah, of course, the leyning and haftarah focus on the Imahos and so do the piyutim in first brachos (the ones that unfortunately many Shuls decide to skip these days but that are printed as not optional in the main text of all machzorim).

Specifically, R' Elazar Hakalir's piyutim in Shachris of the first day (he didn't write the ones for second day), focus on Sarah in first brachah (generally dedicated to Avraham), Rivkah in second and Rachel in third.1

While this last part is slightly off topic, we see 2 things in my response:

R' Elazar Hakalir valued Miriam's role in our prayers for rain (as seen in Hoshanos). R' Elazar Hakalir greatly valued the role of women, specifically the Imahos, and chose to make them, rather than the Avos the focus of his piyutim (within the bracha of Avos!) on the day on which man was created and on which we crown our King.

I would guess that unlike on Rosh Hoshanah, he didn't have a strong reason to bring in a woman to the "Avos" section on Shemini Atzeres where people generally expect to connect their tefillos specifically to to the Avos (defined loosely to include Moshe/Aharon here).

1. The Hebrew text can be found on Sefaria starting here. Select passages from the translation in The Complete Artscroll Machzor are: