You're making this too complicated and that's because you're an IFR student. You have a misconception over how the national airspace system, and instrument flight rules, is built.
Those who learn to fly today are trained in a world where low altitude radar service is the norm. Everything is vectors, RNAV, and directs. While radar is very nice, increasing both efficiency and system capacity, it is a shortcut. It is not the base upon which the IFR system is built.
The system was built based on ground navaids, airways, and pilot nav. The system is there so that you can navigate from takeoff to touchdown without any help from ATC. ATC was added, it wasn't there when we started flying IFR, to provide separation when IFR traffic increased to the point where the "Big Sky Theory" started failing with some dramatic results. That was a time when, if you weren't on an airway or very close to an airport, you were likely in uncontrolled airspace.
When you look at an approach plate, if it doesn't have a note saying that radar is required, it is designed so that you can navigate to it, from the enroute structure, and fly the approach without any navigation help from ATC or RNAV. This is still the fall back if ATC were to go offline (ATC Zero) or during comm failure.
This specific note is there because of the sharp angle of the turn that would be required to make the prohibited transition from the airway to the approach. The radius of that turn would be too wide and would result in you straying outside of the protected airspace.
While under radar contact, the controller can take you outside of the protected airspace, as provided by the approach procedure, using his minimum vectoring altitude by providing both terrain/obstruction separation as well as IFR separation.