Where is my Collatz conjecture proof wrong?

EDIT after OP edited the question.

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Your reasoning fails after point $6$ (included). Indeed, you talk about a certain number in the chain being "smaller" or "bigger" than another, and then mistakenly apply this rule to multiple elements of the chain (for instance in your point $6c$).

This is not strong enough, as $n/2$ might be followed by $3(n/2)+1$, which is always bigger than your original $n$. And then the same behaviour might happen again, etc.

So you have not proved that every $n ≡ 1\ ( mod \ 3)$ eventually becomes smaller than $n$.

Nor have you proved that there does not exist a cycle other than $4\to2\to1\to4$, a problem that your reasoning did not address.

But, we can "compose" any $n \not≡ 0\,(\text{mod}\space 3)$ from smaller number.

Not true. How do we get $11$ from a smaller number? You've only proven that you don't need to consider odd cases of the form $3k+1$, and you've also proven that even $3k+2$ can be 'composed' from a smaller term. You still have $n ≡ 2\,(\text{mod}\space 3)$ (where $n$ is odd) to think about.