the only way for the sample to communicate with the coil is via electromagnetic radiation.
Not necessarily radiation, but via EM field. The time-dependent electromagnetic field is causing induced EMF in the detection coil, but this does not require that the EM field is mostly EM radiation, it can be just non-radiative time-dependent EM field, like we have in AC circuits.
However, by conservation of energy, the only way for a spin to emit an RF photon is for it to undergo a $\beta\to\alpha$ transition with concomitant spontaneous emission (since there is no stimulating radiation).
Forget for now about "spin emitting an RF photon". That is not an appropriate language or model to understand what happens in usual NMR in time. Nobody is "measuring" individual nuclei spin projections in the sample or counting photons in the coil.
The system is not made of spins either in up or down state, which would randomly flip and emit energy $2\mu B$. Instead, if we solved the Schroedinger equation, we would see the nuclear magnetic moments precess in external magnetic field, and many of them quite coherently initially, and less so later when the effect of different environment of each nucleus manifests. These magnetic moments together produce macroscopic time-dependent EM field. This is not like a photon state, it is more like a coherent state. Most spins thus perform continuous motion over states different from both up and down, similar to classical precession with frequency $2\mu B/\hbar$. During this precession, they are together emitting time-dependent EM field with a strong Fourier component at the precession frequency. With time, the system relaxes, and magnetic moments return back to be close to being aligned with external magnetic field.
The photon language works well for getting the precession frequency: if we, for a while, adopt the fiction that the spins flip-flop between the orthogonal states up,down, we can quickly get the precession frequency from the relation $\hbar \omega = E_2 - E_1$. Just like when we want to get emission frequencies of excited atoms (electronic states); the emitting atoms are of course not all either in excited Hamiltonian eigenstates or in the ground state, they are mostly "in between", in superpositions of the eigenstates, but the photon language works when we want to get the natural emission frequencies, instead of the continuous time-based description of the process.
The more correct method is, of course, to solve the Schroedinger equation in time-dependent field, both for magnetic moments of nuclei and for electronic state of atoms in the latter example, and this shows the emissions frequencies are as gotten from the simple formula used in the photon method.
Does that mean that the current generated in the coil is due to the spontaneous emission of the spins in the sample?
Spontaneous emission is a technical term, emission of EM radiation from atoms/molecules that is not proportional to intensity of external radiation, and when the atoms/molecules synchronization is not important (like emission from previously irradiated fluoroscent dye, or thermal emission from hot bodies).
But as I wrote above, the coil does not necessarily detect only radiative EM field, and the sample is quite synchronized initially. So it's more like emitting antenna. But you've got the main idea right: there is emission of time-dependent EM field from the sample, and when there is no external wave, the sample produces it autonomously, or "spontaneously", due to being in unstable motion which creates some macroscopic oscillating currents. It's just that "spontaneous emission" is not the right term here.
It seems like this would reduce the population of the $\beta$ level and lead basically to relaxation
Yes, but. It will contribute to decay of the density matrix element for the excited state. But this does not mean there are two populations, one with spin up, the other with with spin down. Again, mostly the spins are in superposition, and they precess. The decay of the population of the excite state (=element of density matrix) is just due to the spins getting more aligned with the up state (aligned with magnetic field), not because of spin flips.
This is similar to radiation reaction on current in emitting antenna; there, the current acts on itself to damp itself down towards a non-emitting state. Here, the sample and the coil act back on the sample, and get the magnetic moments more aligned, towards the state where they don't emit. See also the Landau-Lishitz-Gibert equation which should take this into account.
- however, maybe we just don't need to account for it since the spontaneous emission is so weak (its strength depends on $\nu^3$ after all) that other relaxation mechanisms completely dominate?
I didn't check which effect is stronger in any particular case, but you are right that radiative damping depends strongly on frequency, like $\omega^3$, thus may be negligible for low enough frequency and high enough other relaxation effects (dephasing, thermal fluctuations in local field).
