It is approximately what you say.
A free neutron decays spontaneously into a proton, electron and neutrino, because decaying provides energy, because the mass of a neutron is higher than the sum of the masses of the decay products.
A free proton does not decay because none of the possible decay modes can produce particles with a lesser mass.
This is the same reason why your body does not fragment spontaneously in separate parts, but some external energy is required for that, e.g. someone wielding a meat cleaver.
The neutrons forming a neutron star are bound together by the gravitational force. When the neutron star has formed, the energy equal to the binding energy has been lost, so the average mass of a neutron in a neutron star, i.e. the mass of the star divided by the number of neutrons, is less than the mass of a free neutron and it is also less than the mass of a free proton and even less than the average mass of a nucleon inside the nucleus with the highest binding energy (iron 56).
Otherwise the star would have remained composed of ordinary nuclei instead of becoming a neutron star.
To extract a free neutron from a neutron star you must provide an energy at least as large as corresponding to the difference in mass between a free neutron and the mass of a neutron bound in the neutron star.
To make it "decay" (of course, that is not decay, because it is not spontaneous) while remaining in the neutron star, you need to provide some lower energy, which could convert a neutron into a proton, electron and neutrino, creating an excited state of the star, like an excited state of a nucleus or atom. Soon after that, the difference in energy will be radiated, either when the proton and electron would recombine again, or the proton will spontaneously decay into a neutron and a positron (which will later annihilate with the electron).
So a neutron star should behave like any other bound system. The state with the lowest energy is the state when all the nucleons are neutrons, unlike the state with the lowest energy of an ordinary nucleus, where a part of the nucleons must be protons.
This being the state with the lowest energy, no decay processes can exist. External energy can produce excited states, where a few protons, electrons and positrons may exist, but these other particles will decay, combine or annihilate, so the base state will be reached again.
The same happens with atomic nuclei, which are bound by strong nuclear forces instead of gravitational forces. The neutrons in stable nuclei or in nuclei with excess protons do not decay. On the contrary, the protons in nuclei with excess protons over the corresponding stable nucleus decay into neutrons and positrons (or they capture electrons).
So a neutron star behaves in the same way as a nucleus where the state with the lowest energy happens to be the one with no protons.