A few thoughts here, responding to a comment where nearly all of the facts stated are wrong and/or ignorant.
Caveat 1: I'm not an aviation lawyer, or indeed licensed to practice law of any kind. Caveat 2: I am a licensed rotorcraft PPL, certified to fly helicopters non-commercially so I have a marginal amount of study of the FAR/AIM.
Now, some background. There are three primary ways to accomplish a flight:
A) Part 91 of the FAR covers non-commercial flights. This includes pleasure flights, such as private aviation, and flight schools. Unless a friend is giving you a ride in their plane (in which case you're only allowed legally to pay your pro-rata portion of flight costs) or if you're a student pilot (in which case your co-pilot must be performing flight instruction and be a Certified Flight Instructor with their Commercial license) you probably have never flown a Part 91 flight. There's really no way that TapJets could have been operating under Part 91: it's not a flight school and it is being run as a commercial operation.
B) Part 135 (not "part 131" in parent comment) covers air taxi / charter operations that are performed on an ad-hoc / on-demand basis. The bar for such flights is much higher than Part 91 flights. Pilots must not only have their commercial certification (250 hours of flight experience for fixed-wing) but also are subject to further training and the craft and overall operations are subject to a higher bar of scrutiny. This is sane: if you pay a commercial provider to ferry you from one place to the next, you have a reasonable expectation of safety.
C) Part 121 "air carrier" operations with regularly scheduled commercial flights. If you've purchased a plane ticket on a carrier (think: United, Delta, American, Southwest, etc) the flight you were on was almost certainly a Part 121 flight. These flights are subject to the highest scrutiny since they have the highest assumed customer safety. Pilots are held to an incredibly high bar, the Airline Transport Pilot level of certification, requiring a minimum of 1500 hours of flight experience. This also makes sense: if you're buying a regular plane ticket on, say, Hipmunk, your assumption is that your flight will be professionally and safely flown.
Now to address the specific commentary:
1) That's correct that if a private pilot is flying their friends and family who are not paying anything for a flight, that is a lawful Part 91 flight. I'm 99% confident this is not the service TapJets is offering.
2) Most commercial planes are not currently certified for single-pilot operations. There has been specific successful effort to do so for Very Light Jets but I would be surprised if TapJets was flying any Single-Pilot IFR certified aircraft. (Happy to learn more here if this guess is wrong.)
3) The FAA's legal jurisdiction is to secure and make safe the navigable airspace of the United States of America. They have full regulatory jurisdiction to make calls on what constitutes safe flight or not and to enforce such rules, as provided by Congress. They are not required to first successfully pursue a civil or criminal trial against an individual or organization in order to enforce regulatory code. In this case it would appear that a Part 135 licensed operator was performing flights outside of Part 135 requirements and has had their Part 135 certification revoked. TapJets "stating" that they conform to Part 135 is not sufficient; the FAA appears to have solid evidence that they have not.
For better or for worse, flight operations in the US are considered a privilege and not a right and Congress has given the FAA permission to regulate things that fly in the navigable airspace. (See US v Causby to begin going down a rabbit hole of the still not-very-well-defined notion of what constitutes "navigable airspace" but it certainly extends at least from 400 feet above ground everywhere over US territory to the edge of space.)
You may be annoyed at FAA regulation but their partnership with the NTSB investigating and remediating every aircraft accident has led to a truly astounding level of safety: ~0.07 fatalities per billion passenger miles. You're more likely to die on a bus, and buses are pretty safe. And compare with 212 fatalities per billion passenger miles on a motorcycle. Eep! [source: https://www.sciencedirect.com/science/article/pii/S073988591... ]
So all that burdensome regulation has produced cheap, plentiful, inarguably safe air transit.
I think the FAA is doing just a fine job.