This is probably a mistaken belief or an excuse to keep visitors off their land. I haven’t seen any scientific data to show that hill walkers have any significant effect on the wellbeing of ground-nesting birds [If you have, please provide a link]. I can understand that rock climbing might upset cliff-nesting birds but I’ve read that ground-nesting birds tend to nest away from paths. Hence signs telling you not to let your dog run off-lead off-path in sensitive areas.
I recall reading a New Scientist article some years ago about the factors affecting the numbers of ground-nesting birds in moorland area. They are far and away due to the presence (or otherwise) of gut parasites, prolonged bad weather and predators. I would imagine walkers to be in the noise level.
The Countryside and Rights of Way Act 2000 (CROW Act) has opened up huge swaths of the UK to walkers (albeit with sensible restrictions). Had that adversely affected the livelihood of upland landowners who get income from grouse shooting, etc., that act would not have come into law.