Is bending stiffness reduction a good model for beam wear and tear?

"Wear and tear" is not a commonly used term in Mechanics, I fear you will not find many references. Might be fatigue, abrasion, softening, ageing, etc. Depending on the underlying physical causes. On the other hand, as long as "wear and tear" happens uniformly along your beam, reducing $EI$ is a perfectly sound approach: indeed, the beam will just bend more, be for loss of cross section, softening, etc, and the effect will be captured by your approximation. In fact, curvature is all there is there, to describe an Euler-Bernoulli beam. The approach will fail if whatever "wear and tear" does not occur uniformly along the length: for example, if the more stressed regions degrade faster, or abrasion occurs predominantly on one portion of the beam. In such case, $EI= EI(x)$ will have to be modelled as a function along the length. Also, care might have to be taken in the numerical solution of your dynamical Euler beam PDE, if the "wear and tear" occurs at a rate comparable to the beam's frequency, but that should not be the case.