What "Los Angeles taxi union" organization specifically lobbied against the LAX train, which, by the way, is now being built?
There may well have been industry lobbyists fighting this but until very recently (the last five years) almost no taxi drivers in Los Angeles were unionized.
The Green Line (opened 1995) was originally supposed to have a real LAX station, instead of the (current) station-near-LAX-with-a-bus-transfer. The Crenshaw Corridor train (which connects to the Green Line) will open an LAX station sometime in 2021, if it isn't delayed.
The Green Line (opened 1995) was originally supposed to have a real LAX station
...until the FAA intervened and said the Metro couldn't build all the way out to the airport, due to the potential danger from crossing existing flight paths. (In a nutshell: the Green Line would have crossed perpendicularly to the existing runways, which would be fine normally but a huge risk in the event a plane overshoots or undershoots the runway.)
There may well have been industry lobbyists fighting this but until very recently (the last five years) almost no taxi drivers in Los Angeles were unionized.