"the fairly-recently-painted stairs have holes in (which he thinks would be painted over if they were old)"
that is probably true.
Have a look underneath the stairs, look for wood dust, frass, holes. Hoover it all, then any new material will be very obvious.
I expect your house is old. Worm don't like modern centrally-heated houses because the wood is too dry. They like warm, undisturbed, dark places, like under the stairs, the bathroom floor, and round the loft hatch where warmth creeps up in winter. If you have bought any second-hand wooden or wicker furniture, worm may come out of it and infest the house.
The house ought to have been inspected for worm each time it was mortgaged in the last 50 years or so, and usually a certificate of treatment is required.
Inspect the loft timbers, because damage here can be very serious. Scrape away loose material and poke the wood with a flat-ended screwdriver (not anything round or pointed as the marks will look like worm holes).
Spraying is not expensive, but it smells and you have the work of removing carpets and other coverings. Parts of Southern counties around Surrey have the Long Horned Beetle which is very destructive and treatment is essential.