Also, it should be pointed out that the inability to hold up against gravity has a lot to do with relativity, and how fast the particles that are responsible for the pressure are moving relative to c. When the electrons go degenerate in the core, they soak up virtually all of the kinetic energy from the ions (degeneracy drives their temperature down, causing heat transfer between ions and electrons that shifts the kinetic energy to the electrons). When the electrons have the kinetic energy, they are also responsible for the pressure, since the pressure comes from kinetic energy density at the micro level. But electrons are low mass particles, so they get moving very fast as they gain kinetic energy, ultimately reaching close to the speed of light (in high-mass cores only). That's when something very important happens-- the dynamical stability shifts from highly stable (nonrelativistic particles) to neutrally stable (highly relativistic particles going close to c). Physically, it is because if the electrons are already going at close to c, they cannot be made to go much faster as gravitational energy is released, and this compromises their ability to push back as the contraction continues.
The situation is not quite dynamically unstable when relativistic, but it's close enough that any process that loses heat (like creation of neutrons, photodissociation of nuclei, and escape of neutrinos) easily makes the core collapse catastrophically. It's all about the electrons approaching c-- degeneracy is only relevant because it robs the ions of their stabilizingly non-relativistic share of the kinetic energy that is going into the pressure. There is no role of "degeneracy pressure," indeed that is not a type of pressure at all, the term merely means the gas pressure that is present, owing to mundane kinetic energy density, when the temperature has been driven down very low as the system approaches its Pauli Exclusion Principle regulated "ground state" .
The same holds when neutronization occurs-- pressure still comes from kinetic energy density, so the "bounce" only happens because when the pressure is held by higher mass neutrons rather than lower mass electrons, the particles are less relativistic and hence more dynamically stable. However, there are still heat-loss mechanisms that would doom the neutrons to collapse, except that the system's PEP-regulated ground state is being approached, and that can cause the temperature to be made low enough that heat loss is inhibited enough that collapse cannot occur. If so, you get a neutron star, but if the heat loss is not inhibited enough to prevent further collapse, you get a black hole.