The CDT is on my list of future hikes as well. But the NM section doesn't appeal to me for the reasons you stated. Basically you're filtering from stock tanks along the way and the distances can be great between them.
I would do the Gila Wilderness section, however. But, of course, you have the Gila River meandering the whole section.
I think I'd start in the South San Juans in CO close to the NM border and hike north from there.
Map and compass and/or
GPS skills are necessary on the CDT because sometimes the trail disappears.