The comment is not true, you are enriching their soil, only the wood takes time to break down and is more visible when on the surface, its best place!
Hi Bessie, good you have fine harvests. The comment about ‘wood taking nitrogen’ is true only when it is incorporated in soil and thus in contact with soil organisms at rooting level, leading to competition for nutrients. On the surface, it may be awkward for sowing carrots for example, and you don’t want too thick a layer because of that, but it takes only a few nutrients where in cantact with the soil surface, and also has access to air bacteria (azotobacter) which help bring nitrogen from the air..
Still tricky for you to handle the owner’s comments, hope it works out. They should be happy when they can see good crops there.