Radio stores are notorious for "fishing" for a hot lead and splicing into it whatever it goes to, and they are also very bad about just slamming things in and screwing it down tight without checking for pinched wires, etc.
It's entirely possible that they either broke something or pulled something loose while bashing around on the radio. They are obligated to take the car to a dealer or indie and get the CC fixed, they damaged it and "there's nothing wrong, we didnt' do it" won't cut it.
If the CC doesn't work, it's broken -- this really isn't a discussable item.
Most likely it's either a cut wire or pulled plug.
The dealer is right, installing a new MERCEDES radio won't affect the climate control, but what about a NON Mercedes hack job? All bets are off! I've got a real hack job to fix on the 280, the idiots cut the loom between the original Becker and the amplifier and "grabbed" a hot wire, leaving the rest bare. Used wire nuts to connect tiny wires to the factory speakes, rear only, etc.
1972 220D ?? miles
1988 300E 200,012
1987 300D Turbo killed 9/25/07, 275,000 miles
1985 Volvo 740 GLE Turobodiesel 218,000
1972 280 SE 4.5 165, 000 - It runs!