I respect you for asking that question and to be hesitant to pull the dash for the obviou$ reasons.
It sounds like it isn't your temp controller because it did adjust the temp. It sounds like a vacuum controller to me because it didn't bring the air out of the center vent, which is controlled via a vacuum actuator. Since your 260E is of the same vintage as the 190e I just did (16 +/-) years old, I would have to guess again that it is the vac mods.
If you want to be conservative and approach the problem very systematically, there are step-by-step flow charts to determine and verify what components are failing, I believe found in the 124 repair CD.
It occurs to me that since all the vacuum lines for this system terminate under the dash on the right side, that one could drop the lower section, find the vacuum control block, find each line that goes to each specfic vac mod and test it for leakage.
Some of the vac mods are single connections (one function) while others have two vacuum connections supplying a regulating feature. My point here is one side could be good while the other has a tear and creates a vacuum to leak (doesn't work). The lines are specific and can be traced.
In a nutshell: If it is determined that you do have leaking vacuum modulator(s), you will have to dig into the dash. Some are much more accessible, the center console area, while others are much more time consuming and require removing a lot of parts. If you are a DIYer it isn't really that bad.
To make you feel better, at least your heater core isn't leaking ....
Hope this helps,
'03 E320 Wagon-Sold
'95 E320 Wagon-Went to Ex
'93 190E 2.6-Wrecked
'91 300E-Went to Ex
'65 911 Coupe (#302580)