Approach towards proving some symmetries of prime numbers

Question

Reviewing draft documents from years ago, I found an interesting line of reasoning for approaching a proof of what I called the "symmetry conjecture" about prime numbers, which can be stated as follows:

For every $n\in\mathbb{N}$, there are at least two pairs of prime numbers $(p_i, q_i)$ and $(p_j, q_j)$ such that 1) $p_i+q_i=2n$, and 2) $p_j-q_j=2n$

The first part of the symmetry conjecture is of course strong Goldbach's conjecture, and the second conjecture I currently do not recall to whom is attributed (if someone knows, please share it!)

To approach a proof of the second conjecture, I devised the following line of reasoning:

1. From Bertrand's postulate, we know that for every $n\in\mathbb{N}$ there exist some prime $p_0$ such that $n<p_0<2n$
2. Let us suppose that there exist some $n\in\mathbb{N}$ such that the second conjecture is false.
3. Let us state that $M_1=p_0+2n$. From the supposition stated in 2., $M_1$ is composite of at least some prime number $p_1$ less than $2n$. We can then state $M_2=p_1+2n$. In turn, from 2., $M_2$ is composite of at least some prime number $p_2$ less than $2n$. And so on...

4. As the process described in 3. continues indefinitely if the second conjecture is false, and that cannot be true since there are only finitely many prime factors less than $2n$, we can conclude that the second conjecture is true.

For the strong Goldbach conjecture, it can be followed the same line of reasoning with slight modifications as follows:

1. From Bertrand's postulate, we know that for every $n\in\mathbb{N}$ there exist some prime $p_0$ such that $n<p_0<2n$
2. Let us suppose that there exist some $n\in\mathbb{N}$ such that the strong Goldbach conjecture is false.
3. Let us state that $R_1=2n-p_0$. From the supposition stated in 2., $R_1$ is composite of at least some prime number $p_1$ less than $2n$. We can then state $R_2=2n-p_1$. In turn, from 2., $R_2$ is composite of at least some prime number $p_2$ less than $2n$. And so on...

4. As the process described in 3. continues indefinitely if the strong Goldbach conjecture is false, and that cannot be true since there are only finitely many prime factors less than $2n$, we can conclude that the strong Goldbach conjecture is true.

The problem with this line of reasoning lies in proving that the processes described would continue indefinitely, or which is the same, that they will not lead to some closed loop; notwithstanding, I found the argument original enough for sharing.

It would be great if you could give some counterexample to the line of reasoning (showing that is not valid because of the existence of closed loops), or some idea to improve the argument to articulate a proof of the symmetry conjecture (or some other interesting results).

Thanks in advance!