I disagree. A hole is only a hole when it has an uninterrupted “edge”.

Drilling “two” holes that touch makes them just one (larger) hole.

As such, if you were to drill the entire pigeon away you haven’t actually created a single hole. There is no pigeon anymore so there is no hole.

If we were to only allow drilling valid holes (holes that have an uninterrupted edge) then you cannot ever drill a whole pigeon away.