This is false to the point of being deliberate FUD.
The LAX people mover is $4 billion because the FAA requires a fuck-ton of care to be taken so that construction and the people-mover itself doesn't interfere with the operation of the airport. It is a 2.25 mile system with 6 stations: the terminals, the rental car facility, the parking lots, the metro station, and two employee-only facilities.
The shuttle you speak of will only operate for 2-3 years, between the parking facility (expected completion 2021) and the terminals, while the Peoplemover is built out to the terminals (expected completion: 2023).
LA taxi drivers had literally no influence over the design of the airport or public transportation to it. I don't understand the bizarre logic required to think that a few hundred people making just above minimum wage have the political leverage to block a multi-billion dollar project.