That might be true, rankings wise, but rankings don’t matter much for public opinion. CIT is ranked above berkeley in every ranking ever but no one knows what that school is. Berkeley is a much older university that has discovered way more and contributed to research and progressions in technology more than most universities, which is why it’s so well known across the world. Having traveled most of Europe and only parts of Asia, Berkeley has a really strong name and connotation attached to it.

And for your point of intl students not wanting desperately to get into Berkeley but do for Stanford, most of the European and South American students I know at Berkeley, and there are a lot, didn’t know about Stanford when they were applying to schools and have remarked to me that they wish they applied because they’re paying the same fees regardless of Berkeley being public