Since Hungarian does not discriminate between the present continuous and the simple present, how come "The kindergarten teacher looks for a girl" may be incorrect?
It isn't incorrect, report it if it's not accepted.
Kindergartener is a proper shortcut for kindergarten teacher and should be accepted.
Out of curiosity, which English speaking country have you heard the word used in? Here in Australia it's certainly not standard; if it was used I'd imagine it would refer to one of the children, not the teacher.
I (American English) have never heard 'kindergartener' used for the teacher, only for the child.
In Texas, Kindergartener stands for the child that is in Kindergarten, not for the teacher.