The existing surveillance radar system isn't necessary for safe IFR operations. There are nonradar procedures. Every controller and every IFR pilot knows them. Once widely used, they became less relevant in the 1970s as radar coverage improved to the point where most of major terminals and the route structure in the U.S. were covered by radar. Outages still take place however and the nonradar procedures are still used.
In essence the nonradar system involves separating aircraft by time, altitude, or route, and relies on periodic position reports from pilots, at standardized locations. The position reports follow a standard sequence and would read something like "United 123 Gopher at 1851Z, 17,000 feet Halfway at 58Z, Rochester next" where "United 123" is the flight identifier, "Gopher" is the location name, 1851Z is the time over the location (possibly a minute or two in the past due to delays in reporting if the radio channel is busy), 17,000 feet is the altitude of the aircraft, "Halfway" is the next reporting point, 58Z is the pilot's estimate of the aircraft's arrival time over that point (in minutes after the hour), and "Rochester" is the subsequent reporting point.
Radar is a great tool. It improves capacity, reduces pilot and controller workload, and allows for random routes. However, it would be a mistake to think that aviation depends on it or on any automated replacement for it.
The technologies that made IFR flight possible were the gyro systems that allowed the aircraft to be controlled without a horizon reference, and radionavigation. Radio systems that permitted pilot-controller communication came next. Radar was first deployed at congested airports to improve capacity, and it was not until much later that route surveillance radar and ubiquitous transponder use became the norm.
