organic chemistry - How to rationalise the protonation state of amino acids

I am learning about amino acids in various $\ce{pH}$ environments and how their groups are protonated or deprotonated in these environments. I don't need to mention a specific acid for my question, I will just use the generic form of one: $$\ce{H2N-CH(R)-COOH}$$

The $\ce{H2N}$ group has a $\mathrm{p}K_a$ of $8$, I believe, and the acid has a $\mathrm{p}K_a$ of $3.1$.

If I drop these into a solution with $\ce{pH} = 12$, the $\ce{COOH}$ will donate it's hydrogen ion, but the $\ce{H2N}$ will not. It will only add another hydrogen ion if it is basic compared to the solution to become $\ce{H3N+}$. Why is it, that it won't donate its hydrogen ion like $\ce{COOH}$, when it is acidic compared to the solution, to become $\ce{HN-}$?