Do you have RPO code C69 (rear air) and/or C36 (rear heat)?
Dumb question, but have the upper rear air vents ever blown hot air? I wonder if yours doesn't have a heater core (no C36 option) and only had the evaporator core. Meaning, the best it can do is vent air when the a/c is off.
Here is the rear heat/air which has mode and temperature actuators that are controlled from the overhead panel.
That no door actuators are visible in your photo makes me think that there is either no heater core or at all or, at the least, you don't have both rear heat and rear a/c since there is no obvious way for air flow to be diverted from one core to the other.
Check the RPO label in the glovebox, see if those C36 and C69 are shown.
Check under the rear wheel well, see if you have a heater supply and return line going through the floor. Likewise, an evaporator supply and return plus a drain fitting for condensate.
If you have heater hoses/piping, see if they are hot.
As for the water valve in your photo, it was provided on Tahoe and gas engine Suburbans to help with a/c performance. The solenoid is energized when the front a/c panel's temperature dial is set to the full cold detent. That then applies vacuum to the water valve, closing off the heater supply and return lines into the heater core to maximize a/c performance.
My two door Tahoe has it, and I have no rear air or heat.
Reach under the valve and you'll feel a small rod and plunger. It's spring loaded--push up and down on it.
When the solenoid is engaged, the plunger is up inside the valve and two butterflies close the valve, stopping coolant from entering the heater core. When de-energized, it's out and the valve is open as shown in the photo above.
The valve only impacts heating performance when the temperature dial is set to the full cold detent, so it's not going to be your issue.
Sorry for rambling, but the Cliffs Notes question is...
Does your Tahoe even have a rear heater core? Is rear heat provided solely through the floor vents under the front seats?