Suppose that we have a uncharged spherical conductor that carries a cavity with a charge $+q$ in it. The problem I am dealing with consists in answering this question: What is the field outside the sphere? Well, in the solution, it is said that, because the conductor carries no net charge and there is an induced charge $-q$ that is on the outer surface of the cavity, a charge $+q$ will distribute uniformly over the surface of the sphere, and will produce a electric field around the sphere. What I don't understand is, how can a sphere with no net charge can produce an electric field? Maybe I should understand no net charge as the charge INSIDE the conductor, which does not include the surface charge?
The distribution of charge is as equally important as net charge. So even a sphere having net zero charge can have a non-zero electric field at a point(the most common example being a dipole).
But in your question the field outside the uncharged shell is due to the system containing the charge($+q$) inside the cavity and the uncharged shell rather than the uncharged shell only. So saying that the field outside is due to an uncharged shell is completely wrong.
If you remove the shell, then the electric field will change(it will not change in the case where the charge $+q$ was initially at the centre of the shell). And similarly if you remove the charge $+q$, the field everywhere will become zero and thus will have changed from its initial value. So clearly, the field is not due to a single component, rather it is "the field of the system".
The reason why you feel that it is counterintuitive is because you are only viewing the uncharged shell as your system.
Consider it like this the charge +q induces a charge -q on the inner surface of the cavity,but since the sphere is neutral or uncharged a charge +q appears on the outer surface of the sphere.Now to find the electric field at a certain point outside the sphere(at a distance say R(>r )from the centre of the sphere).We can find the electric field by applying superposition principle.The electric field produced by the combination(the inner wall of the cavity with charge -q+outer sphere with charge +q+the point charge)is kq/R².