The probability can then get arbitrarily small, meaning that the expected amount of time needed before the probability of having observed a photon would get progressively larger.
My argument above is semi-classical, but it shouldn't change with a full quantum mechanical approach.
> The probability can then get arbitrarily small, meaning that the expected amount of time needed before the probability of having observed a photon would get progressively larger.
Yes, but the probability is never zero, and the expected time is never infinite. So saying "the intensity drops to zero" is never correct.
The point is not that the probability needs to hit zero, it's that it's not correct to say that you receive a quarter of the power as you move twice as far from the source. It's still true that the expected number of photons per second drops by a factor of 4, but it can drop so far as to render the source dark for an appreciable amount of time.
The paradox claims that the sky should appear bright, which I take to mean that a detector should be receiving light from each point in the sky at each moment in time. It does not say that the detector will receive light from each part of the sky at some point, but that you may need to wait a million years before a particular point flickers and that, even then, there's nothing that guarantees that all points will flicker at the same time.
> The paradox claims that the sky should appear bright, which I take to mean that a detector should be receiving light from each point in the sky at each moment in time.
I don't think this is required for the paradox. All that is required is that the average flux of radiation received from the sky as a whole should be constant, and equal, roughly speaking, to the flux corresponding to the surface brightness of a star. That will still be true, under the specified conditions of the paradox (a universe in steady state and infinitely old) even if quantization is taken into account.
> it can drop so far as to render the source dark for an appreciable amount of time
Ah, I see what you mean: yes, the intensity will be 1/4, but because of quantization, you now have to draw a distinction between the time-averaged power (which behaves like the power does in the classical case--more precisely, this would be the expectation value of the power in the quantum case) and the actual power at a given time, which can vary from the average (even to the point of being zero).
My argument above is semi-classical, but it shouldn't change with a full quantum mechanical approach.