In the days of Imperial Airways etc pilots didn't operate "routes" as such but a "line", a single long thin route with many refuelling and night-stops stretching out to South Africa or diverging off to India and the Far East. They were said to be working "on the line".
The expression continued into BOAC and on into the early days of BA, when the expression "routes" was eventually recognised as being a better description.
"Airline" comes from Shipping Line which conveys the same meaning of a single route, which most shipping lines operated in the past.