“Some of those headstones are so old that all the engraving has worn out.”
May be acid rain?
I have done photographs for “Find a Grave” here in east central Illinois. It is common here for limestone headstones to be no longer readable. Granite and some other stones last longer. I found one small cemetery off the road a ways, in a cow pasture in which most of the limestone headstones had been broken off by roaming cattle and sort of stomped into the ground. I uncovered a few, turned over a few and the engraving on each was nearly pristine. I do not remember the most recent burial, but it was many years ago. My guess it that the soil and plant life cover prevented erosion of the limestone surface.
Another somewhat related problem is that the reduction of acid rain has allowed a significant recent increase in the growth of lichens. Gravestones located under trees become encrusted in lichens and become impossible to read. I do not know if the lichen growth has detrimental effects on the rock?
Yes. Lichen breaks down rock.