The old gardener is right IF the wood is incorporated at all, as opposed to sitting on top, where its contact with the soil surface is the only zone of possible nutrient removal to help it break down. Also on top it is exposed to air and azotobacter bacteria which help breakdown by supplying nitrogen.
Wood chips being large take more time to rot than shavings or sawdust so are probably not rotted within a year, and you may want to add compost again, but not good to spread it on top of chips as they are then inside the soil…
I would aim to keep them for pathways. But if they are small chips you may be ok, hard to be precise here.