These were historically the first emissions that were detected and most of the decay modes are of the type.
Beta decay is as a result of a neutron in the nucleus decaying into a proton. It is energetically favourable for the nucleus to make this transformation.
The Helium nucleus is very stable. As you quite rightly pointed out is has a high binding energy per nucleon. It is to do with energy levels in the nucleus which can be compared to the energy levels of orbiting electrons in an atom. Just like electron shells being filled gives very stable atoms - Helium, Neon, Argon etc, so it is true of neutrons and protons. When the first shell is full you form a very stable nucleus composed of two neutrons and two protons which is the Helium nucleus.
So the examples you gave, 2n3p, 3n2p, etc are nowhere near as stable as a Helium nucleus and are not emitted.
The gamma radiation is just a way for a nucleus to get rid of some surplus energy from an excited nucleus just like em radiation being given off when electrons drop down energy levels when an atom is excited.
gamma's are much more energetic than the photons given off from electron transitions in an atom because the spacing between the energy levels in a nucleus is very much greater that of electron energy levels in an atom.
To compound the shells not being full, the 2n3p nucleus is made more unstable because of the extra Coulomb repulsive force as compared with that in the 2n2p nucleus.
So I assume that the probability of the 2n3p surviving to get out of the nucleus is very small.
Although there is not this extra Coulomb repulsive force within a 3n2p nucleus it is again unstable compared with the 2n2p nucleus because of an unclosed shell and so does not survive to get out.
So because of the very high stability of 2n2p that is the nucleus which escapes.