keiko becoming a teacher didn't strike me as strange at all. In many ways, DS9 was like an early pioneer context... they didn't have schools or school boards (or even that many children to attend). So, like early one-room school-houses, if a school was needed and someone was willing to do the teaching, voila - they were a teacher.
And, frankly, at a university-level, things are currently much the same way. Profs don't, by and large, have any teaching training or teaching licenses. Especially in science fields, profs are hired primarily to do research. Then they're thrown into the classroom and expected to know how to teach. They *might* (if they went through a program like my own) have had opportunity to be a teaching assistant for a few years (i.e. to have graded papers and exams)... but many (most?) will have never designed a course from scratch before.
So, Keiko is at least as qualified as many of the profs that teach our students today!