I'm still trying to find a tight upper bound for the quantity $r_{0}(n):=\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\}$.
My idea is that one should have $\sum_{r=1}^{r_{0}(n)}\Lambda(n-r)\Lambda(n+r)\approx r_{0}(n)$. This boils down to $\log(p)\log(q)\approx r_{0}(n)$ with $p=n-r_{0}(n)$ and $q=n+r_{0}(n)$. So we end up solving the equation $\log(n-x)\log(n+x)-x=0$.
It seems this equation as two roots $a\approx -n$ and $b\approx r_{0}(n)$ between which the above expression is positive.
Now computing $J(n)=\int_{a}^{b}(\log(n-t)\log(n+t)-t)dt$ for $n=28$ I get $J\approx 689$. As this number is not that different from $(n-r_{0}(n))(n+r_{0}(n))$ I solved the equation in $y$ $(28-y)(28+y)=689$ whose roots are $\pm\sqrt{95}$.
Dividing $95$ by the known value of $r_{0}(28)=9$ I get a value close to $b$.
So does one have $J(n)\sim (n-\sqrt{br_{0}(n)})(n+\sqrt{br_{0}(n)})$?
Edit: numerically, the quantity $\dfrac{n^2-J(n)}{b}$ for $n=28$ is very close to $\sqrt{r_{0}(n)^{2}+\frac{\pi}{4}}$.