I was eight at the time and I remember the same thing.
I was born after JFK was killed but during the 30th or 35th anniversary there was a TV program that showed extensive footage of the funeral events. I was struck by my emotions when the procession reached the Capitol from the White House and the Congress came out to greet the procession. I still remember the chills and the lump in my throat when they played "Hail to the Chief". Pretty amazing considering that it was an event that occurred so many tears earlier. It's something I felt again during Nixon's funeral. And I left it again today. IMO, there are few more emotional moments than when the band plays "Hail to the Chief" in front of the coffin of a dead president. Very, very moving.