Why is someone who is pedantic called a pedant and not a "pedantician"? If a person working in obstetrics is an obstetrician, why is not a person working with words not a pedantician?
Because the derivations are going in opposite directions.
Pedant comes from Italian pedante, a teacher; in English it means someone who shows off his/her knowledge and constantly tries to "teach" other people, even if they don't need it. (It does not mean someone working with words, although almost everyone on this site has a touch of pedantry now and then.) If you're being pedantic, you are acting like a pedant.
An obstetrician, however, is a person who works in obstetrics. "Obstetrics" comes from Latin obstetrix meaning "midwife"; the male form of obstetrix should be "obstetor", I think, but no such word exists. Instead, we have "obstetrics" (the duties of a midwife) and "obstetrician", one who performs those duties.
The reason there is no such word as "pedantrician" lies in the difference between the word pedantic and obstretics.
"Obstetrics" is a noun, and "pedantic" is an adjective.
"Pedantic" is derived from "pedant", but "obstetician" is derived from "obstetics"
The reason there is no "pedantrician", is because "pedant" is not a study, unlike "obstetrics." No one is an expert or a doctor or practices in "pedant"