It's an interesting link, and very interesting that these regulations are being litigated after decades of acceptance.
The big picture is that the FAA doesn't want the public to confuse the safety record of an airline with somebody flying a puddle-jumper (daily crashes in USA). The flying public just expects to get there without doing a risk evaluation, and puddle-jumpers are far riskier than airlines.
When you rideshare with Uber, there is a question of insurance.
When you rideshare a single-engine piston plane, there is a question of your life.