The amplitudes do become arbitrarily small, and there's nothing at all wrong with this. In fact the exact same thing happens with electromagnetic waves. Sure we have a quantum theory with photons that places limits on how small a packet of energy can be detected, but light can travel across the universe just fine and become as dim as it wants. The intensity of the wave, integrated over the entire wavefront's area, yields the total energy, which is constant.
If you have a quantized theory, all this means is that individual detections will come in discrete packets. If the wave intensity, integrated over your detector's area, is particularly low, all it means is that the number of quanta detected per unit time will be low. The only thing that is quantized (read: discretized) is the energy of the packets, not the energy of the wave itself, the latter being the average energy received over time, whether in packets or continuously.
As a footnote, it is a common misconception that everything in quantum mechanics is quantized. Really the name of the field was poorly chosen. In reality, quantum mechanics is all about linear operators (often non-commuting ones) evolving according to your choice of the wave/diffusion equation. Some of those operators happen to have discrete spectra ("spectrum" means the set of eigenvalues). Discretization is a natural consequence of certain boundary conditions for certain types of problems, but it is by no means the central idea in QM.