Why can't we harness the energy from neutron + protium = deuteron reaction?

Most stable nuclei have a "neutron separation energy" of eight or ten MeV --- that's the minimum cost to produce a neutron, before you factor in any inefficiencies. You can't get ahead spending 10 MeV to free a neutron, having the neutron deposit its 1-2 MeV kinetic energy in some sort of thermalizing moderator so that it can actually capture on hydrogen, and then capturing the 2 MeV photon.

There are some exceptions. For instance, if you put refined uranium in a big pile, neutrons come out of the pile for free. But each fission event releases 200 MeV and only a couple of neutrons that you could use for your purpose; collecting the 4-6 MeV from putting those neutrons on hydrogen is basically rounding error.

There's also the $\rm {}^9Be(\alpha,n\gamma)$ reaction which you mention. However that reaction is not very efficient. In a high-quality AmBe or PuBe source, the number of alphas that undergo $(\alpha,n\gamma)$ on beryllium is about sixty per million. And the americium or plutonium for those sources comes from ... reactor neutrons on uranium. You don't win there.

Let's suppose that you accelerate some helium nuclei onto a beryllium target to set off this $(\alpha,n\gamma)$ reaction. You magically overcome all the technical problems somehow every 3 MeV alpha particle turns into a 10 MeV beryllium disintegration, all of which energy you capture usefully. That's a terrible operating configruation for a power supply, where 30% of the energy (in the wildly optimistic case) has to be renivested in the operation of the power plant. Compare to a campfire (also a pretty inefficient power source) where you find the fuel on the ground, light it with a tiny match, and it keeps you warm for hours.

Your estimation that the reaction creates enough power to power an neutron producting cyclotron is incorrect. It costs far more energy than $2.2~\rm MeV$ per produced neutron, and so even if you could perfectly efficiently extract the produced energy you still wouldn't be producing any net energy. The main problem is that most alpha particles scatter without actually producing a neutron, and so all the energy spent accelerating those alphas is wasted.