A couple things might be going on here.
First, HYR is not a VOR, it is a DME only. Many VORs are being eliminated across the country, some left with a DME only in their place, and additional DMEs are being installed in some areas to fill gaps, as they are far less costly to operate and continue to assist many aircraft as a reference to triangulate position and update their inertial nav. However, DME has no course guidance like a VOR does, just distance. RZN is also only a DME.
Second, in the sim, every radio nav signal is limited to what’s called a standard service volume (SSV), which is various radii from the station, dependent on altitude. Outside of this the signal goes from 100% to 0%.
In real life an SSV is meant to guarantee reception and use within its volume if you are off a published course (airway or terminal procedure). VORs, DME, ILS, etc can often be picked up much farther than an SSV, especially on an airway, it’s just not guaranteed useable off-airway outside of that. The bottom of the SSV is curved, due to the curvature of the earth, obstruction, etc. Here’s a pic, taken straight from the AIM, of the new service volumes (the old ones are slowly being replaced):
Often times, I’ve found that despite the new service volumes, 40nm is the limit of many navaids. I’m just not sure whether the sim does the new SSVs, as it was built on the back of the old system and database. I’m sure someone will be along quickly to correct me on that. 
Third, are you using a third-party nav database? That could clear up some questions as sometimes there are discrepancies between the in-sim and third parties. For instance, the TDS GTN nav database is usually around a year out of date, so it may have stations on there that have since been converted. There could also be a database error (less likely, but possible).
Fourth, just to clarify, your groundspeed is locking up on the DME? It could be you’re flying in and out of service coverage, or there could be some other glitch. Also, groundspeed only really works if you’re flying directly toward or away from the station. If you’re flying tangential to a station, such as on a DME arc, the groundspeed will read close to zero.
Fifth, the discrepancy between distances could accounted for by understanding that DME operates on slant range, meaning the direct distance to the station, accounting for altitude. This is similar to a hypotenuse on a triangle being longer than either of the two sides (I understand the earth isn’t flat, but keeping it simple here). The discrepancy between the horizontal distance to the station over the ground and the slant range is increasingly noticeable as you get close to the station and have any significant altitude above ground. At 30,000’ AGL, directly over the station, for instance, your DME will read about 5.0nm.
However, if you are using the GPS distance to the same fix (rather than tuning the radio), it measures horizontal distance, not slant range to the ground.
Lastly, it could just be a bug. There are some of those still out there.
Hope that helps!