There are many bus routes to airports that I've taken. What is unusual is if the bus service is infrequent or not intended for the airport. Examples include:
MSY New Orleans. They must not want you to take the bus. It doesn't go to the French Quarter but quite a ways walk from it. It also mainly goes to the airport in the early morning and late afternoon.
MLN Melilla, Spanish controlled city in North Africa. There may be a bus but I took a bus to at least 1 mile away then walked.
JFK New York. There's Air Train service to the subway system. However, if you take the much less trafficked bus, it's cheaper.
SDF Louisville. There's infrequent bus service.