I disagree. The renormalizations themselves do not provide good S-matrix elements. One is obliged to consider the IR divergences as well and calculate (at least partially) inclusive cross sections. Only then one obtains something reasonable. In fact, all inclusive cross sections describe inelastic
processes rather than elastic ones. Exact elastic S-matrix elements are equal identically to zero in QED.
Usually they say that IR and UV divergences are of different nature. I agree with it in a very narrow sense: with very massive photons one can
obtain non-zero elastic S-matrix elements. But in my opinion both divergences are removed from QED at one stroke with correct description of interaction physics.