Why is radar required on some FAA instrument approaches?
Radar is most likely required in this case because there is no defined Initial Approach Fix (IAF). 

Also note that in the profile view TIVNE and HORVI have the word RADAR underneath, which means that if you wanted to fly to either of those way points, including a course reversal in the hold, you would still need to be vectored by ATC via RADAR.

This means that ATC needs to give vectors to final or otherwise provide a specific clearance to establish an IFR aircraft on the approach.

There may be other considerations, but that is the main one that I see here.