I know you have to say
Es geht mir gut.
Ich bin gut.
sounds good to me. Can I say that?
Both are grammatically correct, but with a different meaning.
You can say
but this means that you are an awesome boy or that you are good at something... for example, you could use it in the following situation:
You did something and the other person says:
Then you could respond:
So to answer your question: No, it's not a valid answer for "How are you?", but it would be correct as an answer to "How are you at building sand castles?".
"Ich bin gut" is a word for word translation of English "I'm fine/well/good (good probably only in AmE), but it is no idiomatic German. It is never used in this form as a reaction to the question "Wie geht dir's?".
The answer "Ich bin gut." is not a correct German reply to a question like "Wie geht es dir?", because this answer refers to what kind of person one is, not to what kind of state the person is in.
Alternative common answers are:
or simply just:
Nicht nur antwortet man nicht mit "Ich bin gut" auf die Frage "How are you?", man fragt auch nicht "Wie bist Du?" sondern "Wie geht's Dir?"
"Es geht mir gut" ist aber gut und richtig.
Never heard it said. Wouldn't even dream of using it. I can imagine that being extremely steeped into anglicisms might condition someone to think that it is more acceptable that it actually is, but after 21 years spent on the Internet I still haven't reached that point. In case that wasn't clear enough: no, I don't think it is understandable.