It has, but only in American English (presumably due to German immigrants to the US). In the UK we call kids' first school "Primary School". It's definitely possible that a European from another country would translate "kids-first-school" into Kindergarten if they learnt American English, but a brit if I read "kindergarten" I think "either German or US school" and I would expect people from other countries to leave the term untranslated if they were referring to their local schools.
Primary school is 5 - 11, no? That would be Elementary or Grundschule. Kindergarten is for 4 or 5 year olds, like pre-school.
I think you're right. But that only proves that I didn't know what the word meant!