No, the heart of the issue is definitely the dismal state of health care in the US.
Teachers Unions, UAW, and the AFLCIO are pretty much the strongest lobbies in America.
https://www.followthemoney.org/entity-details?eid=37194&defa...
https://www.followthemoney.org/entity-details?eid=133178&def...
List of all sponsors:
https://legiscan.com/CA/sponsors/AB5/2019
As you can see "Labor" is by far the biggest sponsor to the politicans who pushed through this initiative.
I think you are guilty of bad faith reasoning here. First you state that “big labor unions want more power” and your evidence for that claim is “Labor unions push for bills that prevent bad labor practices”.
To help point you in the right direction, why are the United Auto Workers acting in California when there are no unionized auto workers in California? These unions are always seeking to increase there member base because the more members they have the more powerful they are as an entity.
Unions also have a notion of solidarity. What is good for workers in California is usually good for workers in New York. That is why you will often see unions acting across state (and industry) boundaries, in solidarity with other unions and non-union workers.
Entities act in the best interests of themselves.
If unions were truly invested in the best interest of workers instead of the best interest of the union, they would value the desires of the 4/5 of drivers that want independent work as much as the 1/5 of drivers that want to be employees with benefits.
Most likely there is not a single explanation, usually issues like these are more complicated then that. There is no consensus on whether unions are more powerful in the US then elsewhere, I’m sure you can easily find examples—or spin the narrative—to back either case. And finally there is no evidence for some ulterior motive of driver’s unions, nor even a convincing narrative. Quite the contrary there is both evidence and a somewhat convincing narrative for why the ride-share companies would want to keep their workers from unionizing and asking for full-time worker’s benefits.