You are misstating what Senator Howard said during the debate on the 14th. He didn't say "foreigners, aliens, and families of amassadors."
You are injecting an "and" which changes the context. What he actually said was "This will not, of course, include persons born in the United States who are foreigners, aliens, who belong to the families of ambassadors or foreign ministers accredited to the government of the United States, but will include every other class of person."
Foreigners and aliens define the context of families of amabassadors.
It's like saying "apples, pears, members of the fruit family of foods". Apples and pears are inclusive of the group that follow, not in addition to that group.
WW