Current-perpendicular-to-plane (CPP) read sensors having constrained current paths made of lithographically-defined conductive vias with surrounding oxidized metal sublayers, and methods of making the same, are disclosed. In one illustrative example, at least part of a sensor stack structure which includes an electrically conductive spacer layer is formed. A metal (e.g. Ta) sublayer is then deposited over and adjacent the spacer layer, followed by one of an oxidation process, a nitridation process, and an oxynitridation process, to produce an insulator (e.g. TaOx) from the metal sublayer. The metal sublayer deposition and oxidation/nitridation/oxynitridation processes are repeated as necessary to form the insulator with a suitable thickness. Next, a resist structure which exposes one or more portions of the insulator is formed over the insulator. With the resist structure in place, exposed insulator materials are removed by etching to form one or more apertures through the insulator down to the spacer layer. Electrically conductive materials are subsequently deposited within the one or more apertures to form one or more lithographically-defined conductive vias of a current-constraining structure. Advantageously, the lithographically-defined conductive vias increase the current density of the read sensor in the region of the sensing layers to thereby simultaneously increase its resistance and magnetoresistance. With use of the process of oxidation, nitridation, or oxynitridation on each metal sublayer, degradation of the spacer layer is reduced or eliminated such that the desirable soft magnetics of the sensing layers in the read sensor are maintained.