Lack of modern istrumentation, radar transponders, and other equipment which is normally used by modern air traffic control systems.
That's just BS. First of all, the instrumentation to fly an assigned heading and altitude must already be there, otherwise they just wouldn't be certified to fly. Secondly, there's no problem installing a modern radio stack in an old airplane - you can even have it tucked away in the cockpit so it doesn't interfere with the "old style" cockpit look too much. Tons of old airplanes are retrofitted with new avionics all the time. All you really need is the two boxes near the bottom of this instrument panel (retrofitted to a 1940s era Piper Cub with an admittedly quite pimped out instrument retrofit, the original had far fewer instruments, but still at least a compass and an airspeed and altitude indicator).
The primary issue won't be in the corridor itself, it'll be getting into and out of it from shared public airspace
You do know that ATC doesn't control all aircraft, right? Do you know what VFR means? How about class G airspace? Instruments aren't needed for all navigation and many pilots aren't even certified for instrument-only flight (look up "Instrument Rating") - in fact, these pilots are 100% self-reliant. ATC can provide traffic advisories and suggested headings if you ask them, but they don't have to and aren't even obligated to (if ATC is overloaded, VFR traffic gets dropped first).
and avoiding collisions between the aircraft themselves, many of which do not have anti-collision systems or even radars.
Tons of privately owned aircraft do not have TCAS and civilian aircraft radar isn't even intended to and cannot show other aircraft. Still, people manage to survive fly-ins and other large-scale GA gathering events even at uncontrolled airports. The key, that every pilot knows, is: look before you turn, say intentions before you act.