I will be curious to see how this actually plays out.
I've marked up a map showing the relevant boundaries.
The green segment in the center is the 600 acres of private land that was just purchased by DOI.
The red is the BLM-managed North Santa Teresa Wilderness, and the black is the FS-managed Santa Teresa Wilderness.
To the east in yellow is the San Carlos Apache Tribe. Note that this area of tribal land is closed to all
The dark green square is a 40 acre ranch that has a ranch building and other improvements on it, and is also the end of a reasonably developed road from highway 70 (the 13 mile road referenced in the sale link above). The Graham County assessor lists the owner of this property as "USA", while the owner of the 600 acres is still listed as the "Trust for Public Lands" (that's who has agreed to sell it to DOI). The rest of the land between the red and yellow is non-wilderness BLM land.
Assuming that both parcels are now federally owned and will therefore allow recreational access, there is still the matter of getting across the reservation to get there. If the SCAT allowed people to drive through their land to access federal land, then there would already be access to the North Santa Teresa Wilderness. I'm not sure why that would change now?
As it stands, there is still private and leased state trust land south of the ET Ranch parcel which has always blocked easy access to the Santa Teresa Wilderness from the east side. The DOI purchase here makes no change to that access.
Hopefully this is all clarified in the next few months...