Let $X$ be a (smooth projective geometrically connected) variety over a field $k$.

Consider the set Et$(X,k)$ of finite etale covers $Y\to X$ over $k$, with $Y$ geometrically connected over $k$.

Assume Et$(X,k)$ is infinite. Consider the following question:

Does $X$ have a $k$-rational point?

The answer should be negative in general. In fact, I think one can construct a surface with infinitely many etale covers but no rational points by taking the product of two curves $C$ and $D$ over $k$, where $C$ has infinitely many etale covers and a rational point, but $D$ doesn't have any rational points. Then $C\times D$ has no rational points, but infinitely many covers.

What if $X$ is a curve? Is $X(k)$ non-empty?

Note that the converse is true if we consider curves of positive genus. That is, if $X$ is a curve of positive genus over $k$ with a $k$-rational point, then it has infinitely many etale covers.

I'm mainly interested in the characteristic zero case, but comments on the situation in positive characteristic would also be interesting.