Why is the proton the only stable hadron?

Protons, neutrons and in general hadrons are in the quantum mechanical regime and are not elementary. Elementary are the quarks with their positive and negative charges. Here is the main quantum mechanical image of a proton in rough lines

and here is a neutron

Again in rough lines, they are quantum mechanically bound by the potential with the additions of two fundamental forces, the strong QCD force, and the electromagnetic force. In quantum mechanics, as stated also in the comments, systems are stable when they are in the lowest energy state, and this state happens to be the proton.

The title pretty much explains the question, but I've always thought that it'd be a neutron because of its 0 charge.

The neutron "bag" contains two down quarks , and ends up heavier than the proton bag, which has two up, lighter than down , quarks, so the neutron is in a higher energy level than the proton. The picture is more complicated because of QCD , and the hadronic bags contain a large number of virtual gluon and quark antiquark pairs, as illustrated here for the proton, but qualitatively the argument of lowest energy level holds.

A lot of theoretical effort is expended in trying to calculate the mass of the proton in lattice QCD, where all the contributing potentials are taken into account,