Can GeoLabels be controlled?

This required a bit more work than I initially anticipated. To your second question first, per the documentation, GeoRegionValuePlot is quite versatile in what it accepts, and when you are working with a common, queryable propery, you should use the form

GeoRegionValuePlot[enityList -> "property"]

or

GeoRegionValuePlot[EntityClass -> "property"]

as it simplifies what you need to do considerably. So, you could use

GeoRegionValuePlot[
  AdministrativeDivisionData[
    Entity["AdministrativeDivision", {"NewMexico", "UnitedStates"}]
    , 
    "Subdivisions"] -> "PopulationDensity"]

But, I'm partial to the more queryable form

GeoRegionValuePlot[
  Entity["AdministrativeDivision", {_, "NewMexico", "UnitedStates"}}] ->
   "PopulationDensity"]

On to the actual question. According to the GeoLabels documentation, the function form of GeoLabels accepts 4 parameters which are polygon, entity, position, and data. So, we need to use this function:

Function[{polygon, entity, pos, data}, 
 {polygon, Black, Text[StringSplit[CommonName@entity, {",", " County"}][[1]], pos]}
]

which results in

An alternate approach is to truncate the labels. However, this does not center the labels.

nmcounties = 
  AdministrativeDivisionData[
   Entity["AdministrativeDivision", {"NewMexico", "UnitedStates"}], 
   "Subdivisions"];

nmpopdensity = 
  AdministrativeDivisionData[#, "PopulationDensity"] & /@ nmcounties;

Note that you can only need Thread with Rule:

GeoRegionValuePlot[Thread[nmcounties -> nmpopdensity], 
   ColorFunction -> ColorData["LightTemperatureMap"], GeoLabels -> True] /.
  str_String?(StringLength[#] > 34 &) :> StringDrop[str, -34] // Quiet