Proof that there are infinitely many primes of the form $6k+1$. Proof verification

There is a minor issue. Not every prime divisor of your $n$ is necessarily of the form $6k+1$. (Because, as you noted, the theorem with the Légendre-symbol is only valid for $p\geqslant5$.) You would have to argue that $2$ and $3$ do not divide $n$.
This can be fixed by letting $n=(2p_1,\cdots,p_k)^2+3$ as you noted, or alternatively by observing that $x^2\equiv1\pmod8$ for odd $x$, which means your $n$ cannot be a power of $2$, hence has an odd prime divisor.