I'm not sure there's a reason for these particular curves to have no points, but I think people expect in general that a "random" high-genus curve over Q has no points; so in a situation like this one one might guess that the curves have "only the points they have to have." In order to heuristicize about this I suppose you'd start by guessing what proportion of genus-g curves of height at most H had rational points (but already this is a bit sticky, since in large genus curves over Q are presumably concentrated on some unknown proper closed locus in the general-type M_g) and then ask whether this sum converges for the X_{ns}(ell)!
Edit: Also, of course, this is true when Q is replaced by the function field K of a curve over C a finite field, so long as you require that E is not isotrivial. (This hypothesis is something like the exclusion of CM curves in the number field case.) This follows from the fact that the modular curves X parametrizing various level structures have growing genus; X(K) corresponds to maps C -> X, and all such maps are constant once g(X) is large enough. Indeed, a theorem of Poonen shows that the X even have increasing gonality, which means that a yet stronger statement holds in the function field case: for each d, there is an M(K,d) such that: for all extensions K'/K of degree d, all non-isotrivial elliptic curves E/K', and all p > M(K,d), the Galois representation to Aut(E[p]) is surjective.