The routing events documentation here
https://github.com/Turistforeningen/leaflet-routing#lrouting-events
suggests the events are bound to routing.on() where the code binds them to the map object:
https://github.com/Turistforeningen/leaflet-routing/blob/gh-pages/src/L.Routing.Draw.js#L84
Also the documentation says one of the events is "routing:draw-stop" but in the code its "routing:draw-end".
So binding events to the routing object wasn't working, but binding to map does:
map.on('routing:draw-end')