Your post says "No prime divides the new number, so you've just produced a new prime." What, exactly, do you mean by "produced"? You say produced, not "established the existence of", so it makes it sound like you are assuming a constructive argument.
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.
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.