Why do we do shalosh regalim davening on Shmini Atzeres? If I understand correctly, it’s a separate chag to sukkos (for example, we have a new shechianu in kiddush)? And there are only shalosh regalim – Pesach Shavuos and Sukkos
Correct, it is a separate chag in numerous ways, and a “regel of its own” (Succah 48a). The answer to your question though is that the davening is not merely for 3 regalim, but for all of the yomim tovim, (except for Rosh Hashanah and Yom Kippur which are days of din). After all, we do say “Vatitain l’nu… Shemini chag Hoatzeres hazeh”, and we do say it again in Ya’aleh V’yavo, therefore the davening is for all of the Yomim Tovim. The reason it is called 3 regalim is perhaps because there are three times we “go” to the Bais Hamikdash every year. For the first day of Pesach, the first day of Shavuos, and the seven days of Succos together with Shemini Atzeres. Therefore it is called 3 regalim, (3 times) because of the three times we go there.