I don't remember the fat shaming part, probably because as a 12 year old looksist, I would have picked the same companion. The apes were definitely dumber, in charge by brute force and their own feelings of superiority. I see the series as a thought experiment regarding supremacism, who is, versus who ought to run the world. Case in point the supremacism even among the primates (apes, vs chimps vs orangutans), which today reminds me of the jockeying for supremacy amongst the Shiites vs Sunnis.
If it was as racist a movie as some people nowadays think it was, how could it have been extended into a series based on popular demand in modern America?
At the time there was much progress in color-blindness , case in point the "Rat Pack", and the Dean Martin roasts. Mohamed Ali fit the profile of an ape, given his Islamic supremacism, but he maintained a sense of humor, along with his white allies (Howard Cosell) and nobody really talked about his beliefs.