Elipsoid does not posess unbounded geodesics with no self-intersection.
I do not know a conceptual explanation. My explanation is that (due to integrability of the geodesic flow of ellipsoid) we know the geodesic of the ellipsoid, let me shortly describe them.
The typical geodesics viewed as a curve in the tangent bunlde lives on the Liouville torus and is a winding -- periodic or quasiperiodic -- on it. The projection of the Liouville torus to the ellipsoid is a ring (the projection is singular at two lines which project to the boundary circles of the ring and otherwise is the double cover of the interior of the ring. This implies that each such typical geodesic intersects itself.
Consider now ``untypical geodesics'', i.e., those such that their lift to the tangent bundle lies on a singular leaf of the liouville foliation or is a critical circle. The second type are already closed geodesics ( and on the ellipsoid there are at most 3 of such geodesics of the second type).
Now, the last case, i.e. the geodesic lying on a critical leaf are precisely the geodesic passing through 4 umbillic points, and we know that if a geodesic passes an umbilic point of the ellipsoid it passes through infinitely umbilic points infinitely many times which implies it has selfintersections.