> I think teaching is the sole purpose of a university

Cool. This isn't how the word works in practice. More importantly, it isn't how the trustees of the people who gave those universities the money asked for it to be used. (Nor the government or the granting agencies.)

> if an organization only did research but didn't teach I would not say they get to call themselves a university any more

Again, cool. This isn't true in reality. Research universities famously put research first, which is why they can attract top faculty.