To me, and I think many other outsiders, putting a lot of emphasis on the equivalent of grammars-language-automata looks like mathematical naivety. I don't say this to be rude but because you (and Chomsky) claim to be able to interpret the implications of these mathematical results, but I don't think you are doing so correctly. Grammars look like a human (mathematical) invention and not some deep mathematical structure, and these results appear shallow. In the broader context, lots of mechanisms are able to do Turing complete computation.
This doesn't just apply to grammars. There is a huge array of formalisms (e.g. logics, type systems) out there and most just look like the result of someone saying "what if I did this?".
Chomsky's reasoning is that the arbitrary complexity (recursive structure) of human language implies some sort of low level computational engine to do the relevant computations. I don't have an opinion either way. We can see that LSTMs can emulate this kind of logic, but they also make mistakes. Also, I'm not sure that human reasoning is as logical as it might seem. E.g. I read somewhere (lost the reference) that the earliest languages may have lacked the ability to arbitrarily nest clauses. So maybe humans only emulate logical thinking.
As a smart person, I see things a different way. There are certain hobbies that are only appeal to smart people, like pure math. But there are a lot of things that appeal to a broad range of people, like sport, arts, music and literature. I think it's very good to get involved in these things because they bring balance and a way to related to other people. In fact I created a "no programming related hobbies" rule for myself.
You would be amazed how much you can relate to other people, and how little intelligence is an issue, once you have common interests. I remember in my Judo club (which was open to the public but based at the university) a guy who wasn't attending the university commenting that he kept forgetting that a lot of the people in the club were really smart and accomplished (it came up because another guy complained about getting hit in the face, saying he had to present at a conference the next day).
That's not to say intelligence isn't an advantage in these other areas of life. But it's not a big deal or something that creates distance from others. People will appreciate your intelligence just as you appreciate their spontaneity, athleticism, kindness or any other qualities.
This doesn't just apply to grammars. There is a huge array of formalisms (e.g. logics, type systems) out there and most just look like the result of someone saying "what if I did this?".