> Let $A/\mathbb{Q}$ be an abelian variety. Must there be a finite solvable
> extension $K/\mathbb{Q}$ such that $A(K)$ is nontrivial?

This follows from the conjecture that the maximal (pro-)solvable extension $\mathbb{Q}^{\mathrm{sol}}$ of $\mathbb{Q}$ (the field generated over $\mathbb{Q}$ by radicals) is PAC (Pseudo Algebraically Closed - each geometrically irreducible variety has infinitely many points). It even follows from the much weaker conjecture that $\mathbb{Q}^{\mathrm{sol}}$ is large/ample (every smooth variety with a point has infinitely many points).

It is not difficult to see that the answer is positive if $A$ is an elliptic curve, and I guess that for some other families of abelian varieties this can be not too complicated.

Therefore, I will be very happy to see something nonconjectural, which could work for general abelian varieties.