It's not usually noted, but Olber's paradox relies on a false assumption. The argument goes that the night sky is dark and therefore the universe must be finite in extent or duration or both. But the night sky isn't really dark. Olber was just looking at the wrong wavelength. If you look in microwave wavelengths, of course, you see the CMB. The fact that the CMB is out in microwave wavelengths rather than being somewhere where Olber could see it is strong evidence for an expanding universe.
Being right for the wrong reason can be even worse than being wrong. CMB is very relevant to the question of the color of the sky, it just happens to be pre-star material.
