SPOILERS:
I think the only disservice about this movie was the trailer. It emphasized the horror aspect so much that, when the actual movie transitions from horror to more scifi elements, the lack of horror feels like a loss.
The message here was powerful and I think the flashbacks to Gordy's birthday party really convey it well. The scene is horrifying and I immediately wanted to see the chimp as a monster, but then, as I realized the balloons popping were setting him off, I understood that he's not a monster, just a dangerous animal with triggers. Him trying to give the fist bump shows that - Gordy's still there but chimps are wild for a reason.
The parallels with the ufo are obvious. With all the show animals in fact. The only question I have at the end is why did Peele choose to make the ending so cyclical? The chimp was put down because humans tried to tame it with no regard for its triggers and all hell broke loose as a result. The same is true for the ufo, trying to tame it for a sideshow by feeding it horses. And at the end, the ufo dies.
Maybe that's the point - so few people seem to learn in the movie. Jupe tries to do the same thing to the ufo that others tried to do with the chimp. Even the girl mauled by the chimp is there in essentially the same situation but with a ufo. Maybe it's a message that we're cursed to repeat this cycle.
Or maybe not. This time, the ufo caused its own death by attacking rather than being killed. And the main characters don't overreach or forget for a moment what they're dealing with. Maybe that's a hopeful sign, even among the reporters gawking and repeating the same old mistakes.