What effect would adding additional neutrons have on the forces in a nucleus?

If you take any large nucleus and add protons to it, the electrostatic repulsion between them will make the nucleus more unstable, because the electrostatic force between them is more repulsive at a greater distance than the strong force is attractive

So how come if you add more neutrons, which don't have a charge and so there is no electrostatic force, the nucleus still becomes more unstable?

Also why aren't there groups of neutrons bound together, so purely neutron nuclei (because they are neutral, I'm guessing they couldn't form atoms, because of the electrons needed for an atom)