Despite what the article, and perhaps sensationalists, is trying to imply -- the FAA is a rather reasonable organization on the scale of government agencies, and the approach they are taking is to minimize the risks of flying to uninformed or innocent people who may not be aware of all the issues. This is why as much as a pilot is free to joyride across open water or the desert wilderness, they are not free to do that over populated areas that did not consent to the risks of that activity.
When someone starts acting as a provider of transportation to people they do not know other than for the purpose of the transaction, you start to get far more into the realm of people who sign up to purchase a service where there are not fully aware of the risks. Consider what knowledge you have about entering a friend's car, or a family member's car, versus a taxi driver's.
The philosophy is that, ok, private pilots have trained for this activity, and take on the risk themselves. If they share a ride (and split the costs) under the currently allowed rules with friends/family, those people tend to know the risks as well. And that is a relatively small set of potential passengers who could potentially engage in this activity.
When people start advertising to the broader public that they're available for flights, you start to get people who are unaware of the risks. And pilots who will engage in flights (each of which carries some incremental risk) that would not taken place otherwise. And that is the problem, considering that the FAA is mindful a certain acceptable level of flight activity and risk percentage.
The FAA is not being overly heavy handed in this matter. For all the semi-justified concern about Uber insurance requirements and background checks for cars -- for aircraft and pilots I would hold the bar at least 10x higher.