Does this therefore mean that he the shoitah is not a yid?
There is a saying along the lines that a yid is a "chaylek eloka mima'al mamash".
How does this "sit" with shoiteh is poter from mitzvahs .
A Shoteh is definitely a Jew for all matters, and we must be Mechalel Shabbos to save his life. He is Patur from Mitzvos since he does not have Da'as -- this exemption is similar to the exemption of one who has no arms and is therefore Patur from the Mitzvah of Tefilin. The Pri Megadim (see also Chasam Sofer) rules that one may not feed an Isur to a Shoteh.
Dov Zupnik