Like your Dad, I am something of a fanatic when it comes to this show. Let's face it: it WAS funny, given the period. (And, FYI, I'm 71).
Of course, black people, Asian people, Native American people, brown people....simply did not exist in the minds and real realities of most white viewers of that show. Racial segregation in every aspect of life was rife, complete, thorough, expected and practiced -- both in the South and the supposedly more "liberal" North. (See "redlining" in Chicago).
Most of us who watched that show know that the real star was Don Knots ("Barney Fife") until he left after...how many?...seasons.
Finally, it wasn't just Andy Griffith. How many black (or so-called "minority") people did we EVER see in any of these sit-coms? "I Love Lucy," "Our Miss Brooks," "The Honeymooners," "Father Knows Best," "I Married Jane," "The Bob Cummings Show"? I could go on...and on...and on.