The coordination problem isn't the radar net itself. The problems start with the tactics that were developed to make best use of the radar net.
*Any* coordinated air battle is a very complex thing, and the more players there are in the game, the more complex it gets. I don't have any hard-and-fast data on how the complexity escalates (I'm sure it exists, and is probably so classified God needs to sign for it), but I'm betting the curve is geometric or worse. Coordinating multiple units of aircraft, usually of different types (and, in World War II, different models within the same type), all with their own capabilities, is still one of the more difficult things in the military playbook. Take a look at the US Navy's problems with coordination at Midway (all aircraft form one service, and sometimes from the one ship, and there were still issues), or (for the more modern version) the blue-on-blue problems in Operation Desert Storm. Now, add in the extra layer of complication that comes from private squadrons that use their own radio frequencies, have their own doctrine, and may be flying aircraft whose performance is wildly different from the rest of the defensive force. If that mental image doesn't have you reaching for the Johnny Walker, your nerves are better than mine!
There's also a turf issue that will inevitably crop up. Part of the reason that the British (and later) air defense networks worked was that they allowed concentration of force. A fighter unit might be scrambled to intercept enemy units that were nowhere near their base of operations, based on need. What happens if the Air Defense Coordinator contacts Private Squadron 008, and instructs them to go to Area "A", and they refuse, because their factory is in Area "C" ?
Third, there's the cost issue...by World War II, the cost of developing, building, and operating combat aircraft was already high enough that no company was going to offset the cost of a private air force with tax breaks. By the time the Jet Age is in full swing, the cost of a private air force will have reached the point where you (as the CEO) can either afford to have a company, or an air force, but not both.