For the record, I am a DIY hunter and have never used a guide except in those few areas which require a guide. NM has a problem with reserving 12% (now 10%) of their tags to hunters who hire a guide. These can be resident or NR hunters. I understand the unique land grant status of NM, and recognize the number of extremely large private properties in the state. Few other states aside from Texas have this many large private properties where hunting is largely conducted by guides. This makes NM different than WY and ID, for instance. It also increases the pressure on the remaining public lands for the resident hunters. It is extremely difficult for the resident hunter in NM to draw a tag regularly, just like it is tough to draw a public land tag in AZ or Texas.
The current move to further restict NR's who do not use a guide is not progress, however. It will make very little difference in resident draw odds, but have a staggering effect on the economy of many businesses and ultimately lead to much higher tag prices for residents. The game and fish tried to make up this budget shortfall by including a nonrefundable license fee for NR to apply. It is a downward spiral where all loose.
The original post realted to whether you have a right to hunt BLM or NF lands is not right the question. There is a longstanding practice of charging NR more for hunting tags in the west than residents. Most states have settled on tag fees roughly 5x to 10x the cost of resident tags. I have little issue with this practice, unless it exceeds 10x.
I would propose putting all applications in the same pool and drawing names randomly, regardless of resident or NR, or at least with minimal preference. If this was done universally it would be nearly a free market approach. Are residents in NM willing to spend ~$800 for a quality elk tag? I doubt it. Think about the revenue to the state if market pricing was equal for all of these tags.

I acknowledge some problems with this scenario, but you get the idea. States need to stop conferring "special" status to various hunters. I'm not holding my breath for this approach, given the politicians involved, however.