If that were possible, it would allow an operator to control the drone from a very remote place, and without resorting to satellite radio links, as military drones do...
The simple fact is that the cell network is not advantageous for drone control at least not the kind that people prefer. The architecture of the network by nature allows the signal to drop out bandwidth from tower to tower is not guaranteed. This would allow a situation where a drone flying along would get handed off to a tower that has no available bandwidth and the control signal drops out. Dedicated satellite networks allow point to point communication as well as assured bandwidth when its needed.
In theory yes. However the drone is going to be a bit hamstrung for several reasons.
First, it will be dependent on cell reception to operate. This is going to restrict its operation to populated areas with good cell networks as well as restriction operations to low altitudes, say less than 2000 ft AGL. This makes the drone of limited use for commercial or military applications.
Second either the drone would have to be designed to be cross platform compatible with multiple types of cell networks eg GSM, CDMA, etc. and would impose even more restrictions upon where the craft could be operated.
This would leave the only practical users as civilians operating local within a specific country and most are subject to restrictions such as the United States' CFR Title 14 Part 107.31, which restricts operation of small drones to flight within visual line of sight operations without special waivers.
For these reasons a cell linked UAS would effectively be impractical.