Sitting "on" a chair and "in" a chair are both grammatically correct and should both be accepted. They are a bit different in connotation: one might sit "on" a hard wooden chair, but "in" an upholstered armchair. Also, sitting is not ordinarily reflexive in English. "He sat himself down" (never "he has sat himself down") would imply a deliberate act, perhaps balancing on an unsteady chair for instance, or making a formal entrance, as in "the Chairman came in and sat himself at the head of the table."
Ah, yes, I really don't know how I mixed those up. You indeed helped, have a lingot!
So I guess the most plausible answer to the originally posed question is that; in English we never use the word onto along side the action to sit regardless of movement.... At least I've never heard it that way.
I don't seem to understand when I should use 'sich' and when demonstrative pronoun like mich, dich, etc. Edit: I found a link with a table of reflexive pronouns: https://www.lsa.umich.edu/german/hmr/Grammatik/Reflexiv/Reflexiv.html#Table