But, as you noted, sīdō does also have the perfect sīdī. This is a secondary form made from the present stem. Besides, for sĕdeō we have inscriptionally even the perfect sēsī (in 3.pl.), as mentioned in the OLD.
Sēdō is factitive-causative for sĕdeō. Sīdō is reduplicated from sĕdeō with zero-grade on the stem (i.e. *si-sdō; -sd- will lengthen the preceding vowel as the s disappears).
And it goes on. For sĕdeō we have the present sedit (apparently in good manuscripts) in Venantius Fortunatus and the imperative sedito in lex tabulae Heracleensis (Iulia municipalis) from ca. 45 BCE.
*Not everyone will accept the passive perfect participle sessum for sīdō but attach all of them to sĕdeō, leaving this place blank for sīdō.