If the Planck constant tends to zero, it's obviously just getting smaller, so how would this describe classical physics?

Is this a way of explaining it?:

If energy spacing E = hv, and h --> 0, then E --> 0, which is essentially a continuous energy spectrum and quantization and discreteness goes away. Hence classical physics.

It's about Planck's constant relative to the scale you're dealing with. If your scale is meters, Planck's constant is negligible and you get classical physics. If your scale is angstroms or smaller, Planck's constant is quite relevant and you get QM.

It's not unlike the relationship between SR and the speed of light. At low speeds, the speed of light is practically infinite, so you needn't worry about relativistic calculations. At speeds of, say, .8c, the speed of light becomes important.

That's definitely one way of looking at it. More accurately, you would say delta_E = hv, and so as h->0, delta_E = dE, and you get a continuous energy spectrum that would require us to consider things like intensity of the wave which we've since largely discarded.