Depends on the household.
Al Gore's father was a Senator from Tennessee in the 1930s who helped to craft the Social Security Act and the Tennessee Valley Authority. He was there until 1970, I think. The Gore family was quite prominent. But the song is not about him. He did serve in Vietnam.
It was well known through the draft years and lottery years that the sons of the Senators and Congressmen and anyone else well connected did not serve. If they did serve, few saw any action. Al Gore did serve and go to Vietnam as I said.
I read the link in the OP, but I always thought the song was generic as applied to most well-connected sons.
I said I thought it was about Ted Kennedy because his oldest brother was killed in WWII and his older brother, John, certainly saw action also(PT-109). Ted spent two years in the Army at the American Embassy in Paris, and was discharged as a private; he never distinguished himself. Ted's father was U.S. Ambassador to England before he was born.
Look at the political landscape now and try to guess who will run in 2032. I remember when Mitt Romney's father ran for president, but then I am old. I remember when all three Kennedys ran for president and I hoped to see John Jr. run. I remember when both of the California Governor Browns ran for president; Governor Jerry Brown was of draft age during Vietnam (born 1938) but he became a seminarian, briefly.
MMM, where were Mitt Romney, Newt Gingrich, Tom Tancredo, Duncan Hunter, Dennis Kucinich, Joe Biden and Rudy Guiliani during the draft? John Edwards, Michael Huckabee, and Sam Brownback were not old enough as the draft ended before they were 18.