Honestly, I believe that Keiko's becoming the teacher on DS9 was, at least partly, done in tribute to the old Westerns, where there's this one, single schoolmarm who takes it upon herself to educate an entire small town. Her degrees were never in question; never came up. The community would just pitch in to get her whatever she needed, up to and including the actual building of a small school.
In Real World terms, it's true, there's no logic to Keiko's sudden, arbitrary decision, and even less in Sisko's staunch support of it. Even to the point of making damn sure Jake went. And seeking a qualified teacher was never brought up. Keiko wanted the job, might as well give it to her ... and that's that. But I'm sure some quality control guy in the scripting process pointed out all this and it was soundly rejected in favour of getting on with it.