It's far better to compost agri waste directly than this pigeon-shit(chewing a dollar to shit a penny). Pigeons eat mostly seeds so does sparrows. Any insect/pest eating bird species fare better here wrt farming.
Besides the production output is probably far higher than what the pigeon consumption (not a penny for a dollar as you suggest), otherwise why would these people bother for thousands of years.
Anyway, it seems like the system is much more passive than folks are realizing. The bird waste collects over long periods of time, and is presumably removed in a mostly composted state.