If two jobs are equivalent, workers are going to pick whichever one pays better. That means if Walmart is offering $10 to employees and Target tries to offer $10 to contractors (who have to pay more of the taxes), workers will take jobs at Walmart because they get a bigger paycheck. Target will have to pay more if they want to get people to work there as contractors by an amount equivalent to the difference in taxes.

Which means saying that the company pays this amount and the worker pays that amount doesn't really mean anything. In the case where the worker nominally pays less of the tax, the company doesn't have to pay as much to the worker to make the job equally attractive. It doesn't really change who pays.

Of course this is theoretical and real systems have friction and stupidity in them, so people who are lazy or bad at math may end up taking a worse deal because it was framed in a particular way. But that's an information problem -- it has information solutions.

Instead of spending all this time campaigning to classify people as employees, just send out pamphlets to all the Uber drivers with a cost breakdown of what the equivalent employee wage would be compared to what Uber is paying them. Maybe some of them will quit and then Uber will have to pay the others more. Maybe the people doing the job aren't as dumb as they're made out to be and have their reasons. Either way, you solve the information problem with information, not prohibition.