There was a time, perhaps when our better sensibilities knew it, when we were allowed to laugh at all races, religions and socio-ethnicities without being labeled "racist". It was a gentler, more naive time indeed, and the airwaves broadcast TV aimed at the silly side of life. "Gilligan's Island", "The Beverly Hillbillies", "I Dream of Jeannie", "Petticoat Junction", "It's About Time" and of course, "F-Troop".
There is a common thread in all of these shows: Simple, honest people are ennobled. Officious, pompous people are made fun of. Everyone is fodder for fun - no-one is above being poked at.
Ken Berry as William Parmenter is amazing in his comic timing (Mayberry RFD was a big step down for me). Melody Patterson is absolutely delicious jail-bait as "Calamity" Jane, and of course Frank DeKova and Don Diamond as Chief Wild Eagle and Crazy Cat, and Forrest Tucker and Larry Storch, as Sgt. O'Rourke and Corporal Agarn, respectively, are mirror-images of avarice and opportunity.
The relationships of these last 4 characters were the most typical of TV, but smartly turned on it's head: Agarn and Crazy Cat, full of ideas and energy, scheming and snatching at everything that moved, in their climb to "success". Sgt. O'Rourke and Chief Wild Eagle, as the "Establishment", wisely knowing when to take opportunities, but at the same time wringing their hands about their underlings almost as to say "What is it with the kids these days"?
This was wonderful social satire loaded with sight-gags, something for young and old. Unfortunately we Americans seem to have lost the knack for subtle comedy, as we now linger under the thumb of blistering insults and mechanical obviousness. I don't know if we get it ourselves these days - perhaps that is why people look at the show and react first without giving the show any thought.
I don't mean to discount the valid views of other, more PC posters, but they're missing the point. TV and film are just time capsules...you can no more examine history through something like "F-Troop" than experience the future through something like "2001". Ultimately, they're both the '60s.
What you can do is understand the period and sensibilities of that time, and remember one major lesson - something we were learning then but have perhaps since forgotten: That we are all the same under the skin. And at best, we should be taken very, very lightly.