It is possible that the OP did not make the mistake that you are attributing to them, although best case is that the exposition was simply unclear.
But that's not how I argued. Instead I showed that 15 is the new prime because it's relatively prime to all the primes. (By the assumption we had started from all the primes.) And this is what falsifies the assumption.
I think my argument is fine and I don't need to address whether 15 is really a prime.
"Any number that is relatively prime to all other primes is itself a prime"
You would need to first state and prove this in order for your original post to constitute a rigorous proof.
Do you mean that the number you've produced is prime? Or do you mean that "the number you have computed is either prime or has a prime factors not in your list of primes, therefore, by computing the factorization of that number, you produce a new prime"?
The former statement is demonstrably false, the latter statement is true. Others have read your proof as stating the former, and I argue that the latter is a generous but not unreasonable interpretation of your phrasing.
Remember that we are arguing from the false assumption that you start with the finite set of all primes, up to some largest prime, L. Call this false assumption A. It's fine to make false statements that follow from A, no problem at all. Indeed this is the point.
One definition of a prime is that there are no primes smaller than it in its prime factorization. We'll call this definition NSPIPF - no smaller prime in prime factorization. Is NSPIPF an OK test for primality? Sure.
So when you get to new_number, produce the prime factorization. Does it meet the definition of NSPIPF? Yes, because we made it relatively prime to every prime (under assumption A).
It passes primality test NSPIPF. Under A. And therefore is prime. Under A. (This is the part you and others object to, but it's absolutely flawless application of the NSPIPF primality test.) It doesn't matter that in some other way I could also get to a contradiction. For now this is what we do.
Under A, using NSPIPF primality test, we just proved new_number is prime.
Next we show that this new_number definitely wasn't in the set of all primes, since it's larger than L, having had L and a bunch of other positive integers as factors and then one added to that for good measure. It is at this point that we show explicitly that A cannot be true, because we used A to produce a prime that wasn't in the set of all primes.
It doesn't matter if that was a false statement!
I think all this is in my original comment and it is a flawless proof by contradiction. I asked you to "Suppose there are just finite primes, up to some largest" and you gave up when you started seeing false statements, rather than at the end of the paragraph where I presented the conclusion in black and white.
