When you use modifiers on a curve, there are two different things you could be modifying: the mesh, which is ultimately generated by the curve modifier, from its geometry settings; or the spline itself, which is used for things like curve modifiers, follow path constraints, etc. The default settings for curve modifiers set them to do the first, affect the mesh rather than the spline.
Some modifiers can only affect the mesh. But the shrinkwrap modifier, which you're using, can affect the curve instead. How do you make it do that? By clicking on the button to the right of the modifier's name, the button with the tooltip "apply on splines". (For some reason, this button is in a different place in your screenshot than it is for mine.)
Once you hit that button, the shrinkwrap will stop directly affecting the curve's ultimate mesh, which you don't care about, and affect the control points and handles instead. One of the things this might mean is that because the position of the first control point moves relative to the origin, that your curve-modified mesh is no longer in the correct position. The best setup for a curve modified mesh doing this is, origin of curve at its first control; curve and modified mesh object with exact same world-space transform; both curve and object parented to the same object, for control purposes (like an empty.) Set up like that, you won't get problems from the difference between the origin of the curve and its modified position. (An alternative is to use both a shrinkwrap constraint and a shrinkwrap modifier on the curve, which will prevent the first control from moving off of the curve's origin.)