Let $K$ be a field, $\overline{K}$ an algebraic closure, and $X$ be a finite type $K$-scheme.

Could there exist a proper subfield $L\subset\overline{K}$ such that the natural inclusion $$X(L)\hookrightarrow X(\overline{K})$$ is surjective?

If so, what sorts of conditions can we put on $X$ to ensure that this can't happen? I'm happy to assume that $K$ has characteristic 0.

Let $K$ be a number field, $\overline{K}$ an algebraic closure, and $X$ be a positive dimensional finite type $K$-scheme.

Could there exist a proper subfield $L\subset\overline{K}$ such that the natural inclusion $$X(L)\hookrightarrow X(\overline{K})$$ is surjective?

If so, what sorts of conditions can we put on $X$ to ensure that this can't happen? I'm also curious about the answer for more general fields $K$.