As others have pointed out, Express Bourricot will not pick you up at Bilbao airport. They are located in France and will drive you from
Bilbao in Spain but not the other way round. However, their price for the journey, around €300, is most likely a good indicator of what you will have to pay for a Spanish taxi from Bilbao to SJPP.
When you travel from Paris Montparnasse to Bayonne you will be spending 4 hours on the train. These trains are fast
Plenty of people fly to Biarritz, Bilbao and Bordeaux and then use train, bus or shared taxi service to get to their Camino starting point. Your question probably refers to travellers from the USA. These small airports have no direct flights from airports in the USA. They are served by a limited number of airlines. Choices for a flight from the USA with one stop and reasonable transit and total flying times may be limited: for example, there may be attractive flights from an US airport to Madrid but no direct connecting flights from Madrid to Biarritz. Flights from Frankfurt in Germany may be available only a few days per week. Flights from Dublin in Ireland may require a change from a US or code sharing airline to low budget Ryanair with the disadvantage this may bring (think of delays!); same for Paris, should you chose to continue on a flight to Biarritz on EasyJet. Changing planes at Heathrow in the UK may, for whatever reason, not be as attractive as changing planes in an EU/Schengen airport or simply look like too much hassle ...
I've flown to Bordeaux, Bilbao and Biarritz to get to the Camino but I live in Europe. These days, though, if I were to do it again, I'd travel to Bordeaux and Biarritz by train. Flying to Paris and then taking the high-speed train to Bayonne appears to be an attractive option for numerous pilgrim-travellers from the USA.