Seems like a DME without a VOR would be of limited use, especially in today’s MON era.
You're thinking about this from a G.A. viewpoint. The National Airspace System (NAS) is designed primarily for commercial aviation with G.A. as an additional user.
Large airplanes have inertial navigation systems that provide data to their flight management systems. They'll have at least two inertial reference units, many have three. The IRUs can dead reckon, from an initial position when aligned before takeoff, and maintain a fairly accurate position without external input. The IRUs also provide attitude references for the flight instrumentation systems.
The FMS position(s) would drift over time from IRU precession. Various systems have been used to update the position including manually entering a position when flying over a VHF fix. Most commonly, with a system known as DME/DME/IRU. That used DME fixes, from multiple stations, to update the FMS position by tracking the amount of drift in the IRS position. For that to work, the FMS had to have a current database of the area in which it was flying and there must be ground stations with good geometry within range. When you fly out of range of the ground stations, or out of the limits of your onboard database, DME updating wasn't possible and the FMS would dead reckon from the IRS data.
The 767, without GPS for example, could cross an ocean for up to six hours without any position updating other than the IRS. At the end of the crossing, it might be a mile or two off course due to map shift but the separation standards for oceanic airspace, at that time, allowed for that and position updating would resume as VHF ground stations came within range. Today, heavily traveled oceanic airspace require higher accuracy.
When GPS came along it provided a much more accurate and reliable method for updating the FMS(s) position(s). Normally, the GPS updating is the most accurate and that is what the FMS(s) use. When GPS isn't available, DME updating is used. If both are unavailable, the system degrades to dead reckoning.
The accuracy of your position(s) is displaced as an Actual Navigation Performance (ANP) figure. With GPS updating, it might be 0.2 or 0.3. The type of airspace, or type of procedure, you're flying will have a Required Navigation Performance (RNP) requirement. A standard RNAV approach will have an RNP of 0.30. RNAV RNP approaches can go as low as RNP 0.10. Oceanic airspace can be either RNP 10.0 or RNP 4.0. Domestic enroute is RNP 2.0 and terminal (App/Dep) is RNP 1.0. If your ANP falls below your current ANP (a higher number) you'll get an alert similarly to how a bad VOR signal will produce a flag.
You'll see some RNAV procedures with the note "DME/DME/IRU - N/A". That means that even if your non-GPS FMS is achieving an otherwise sufficient RNP for the procedure, you are not authorized to fly it with only DME/DME/IRU.
That is why the DME only stations are valuable in the NAS. They provide a fallback for position updating when reliable GPS is lost.
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