It's been over 30 years since my last detailed US History class, but IIRC, that clause did apply to those born before 1787: had to have been born in one of the 13 by-then-ex-Colonies. So, yes, Hamilton was excluded. I believe the thinking was that local royalists would be known, having already declared their sympathies a decade previous (and in most cases already departed).

As to why put in the Constitution, that was the only available vehicle at the time they were considering the issue -- and they considered it a pressing one.