I have the basic idea: for small nuclei the mass of product is less than the mass of the individual pieces, this mass is released as energy via $E=mc^2$. But why do nuclei use the mass for energy? Versus… why not lose some Kinetic Energy instead, or some other kind of energy? Is there some reason, and is there other examples other than fusion/fission where mass is released as energy?
It's not that the nuclei use mass for energy, they release energy in the process so by definition their mass decreases. Its something that always happens: If you run you're more massive than if you just stand still. The same for a hot and cold gas. Energy contributes to the mass, so losing energy meaning losing some mass. This effect is negligible most of the time, but the nucleus is so tiny and so much energy is released in nuclear processes that it isn't a negligible phenomenon.
E=mc^2 is a confusing term, the m is the inertial mass and is a function of the velocity and it has a confusing algebraic relation to what is really happening.
In special relativity individual particles have what is called an "invariant mass" and it is the mass in the table of elements, and the mass of the proton and the mass of the neutron.
Invariant mass is invariant to Lorenz transformations.
Because of the quantum mechanical binding of nucleons within the nucleus, there is a binding energy per nucleon which comes from the difference between the sum of the invariant masses of free nucleons ( neutrons and protons) and the invariant mass of the nuclei, which is what keeps them stable.It is called the nuclear binding energy .
For fusion , when low binding energy nuclei join up to form higher binding energy particles , a nuclear reaction happens which takes away the excess energy as kinetic energy of new particles and nuclei.
Deterium + Tritium gives an Alpha particle plus a Neutron.
which by energy and momentum balance, share the released energy as kinetic energy: ( 3.5 MeV the alpha ) ( 14.1 MeV the neutron) respectively.
Just forget e=mc^2 unless you are dealing with inertial masses.