Like Billman, I don't reckon the chainsaw had anything to do with the demise of the billhook. Except in vineyards, the billhook was vanishing from the tool box decades before power saws of any kind. By the last quarter of the 19th Century, about the only use for billhooks was military. The army used them for building defences on the battle fields, primarily for gabions and fascines. Military manuals, in fact, call them "fascine knives" from their primary purpose. The military seems to have replaced them with machetes, doubtless because the billhook is a harvesting tool, and when sandbags came into primary status for field barricades, all they needed was clearing tools since harvesting small diameter green wood lost its importance.
Outside of making fascines (bundles of straight, green wood bound together) they are most useful for making hurdles, and land clearing for slash and burn agriculture. We long ago stopped making hurdles, and with our established farms, no longer needed to clear virgin land.
Smaller versions of the billhook (5 to 8 inch blades) continued to be used in viticulture, and may still be used -- not sure here -- but still into the 20th Century.