There's a bit of confusion here regarding the term overlay. In my case I am using it to refer to the fact that the FMS is flying what is coded in the box, and that is the DME arc in my case or whatever other procedure you are supposed to be flying. In reality my chart simply says ILS VOR DME it is a traditional approach. I did my flight training Canada so I know that there are certain procedures known as overlays. I am not referring to that here. What I mean is that we are essentially flying our own overlay, and we have become so confident in the FMS and with GPS navigation that we happily let the computer fly the procedure oblivious to the NAVAIDS we should really be monitoring.
The airfield we were going into had no radar. They could not have given us radar vectors, all their approaches are based off using the VOR and the DME to set yourself up for final. In reality the airport should really not have been accepting any IFR approaches. When we informed approach of the issue their response was just OK, we'll let someone know about it and we just happily continued along, confident that the FMS would get us to the IAF.
The other issue is that a DME arc is a curve, and an FMS is not meant to fly a curved path based on its own internal position estimates except during RNP-AR (SAAAR) approaches, all other RNAV legs whether terminal or enroute are straight lines.
Ultimately I believe that legally speaking what we did was wrong, and that it speaks of our age that we continued oblivious to the obvious.