The system was designed to be used at ground level, with cell phone towers usually about 150' in the air, so that at most a cell phone could only 'see' towers within a couple of miles. Raising the cell phone to 5,000' means you can see cell phone towers tens of miles away.
Usually it doesn't matter, and it's done all the time. (The heroes of Flight 93 on 9/11 used cell phones to call loved ones and that is how they found out the fate of the other hijacked aircraft.)
Still, it's illegal to do so, and I wouldn't be admitting to doing it in print.
They could be (probably are) using dedicated air-to-ground freqencies. From your FCC link....
The FCC has approved rules that allow in-flight voice and data services, including broadband services using dedicated air-to-ground frequencies that were previously used for seat-back telephone service.