When you turn the key to preglow and the buzzer stops, do you hear the aspirator motor whirring behind the glovebox? It should be on regardless of the function of the HVAC system. Even when it is "off" it is still running with the key on.

The next thing I'd suspect would be the pushbutton unit. Since none of the functions work, including blower, that would be getting the stink-eye from me. Mine kept dropping the fan out when hitting a bump in the road, fiddle with the fan switch and it would come back on. When I took it apart, nearly every switch in the thing had broken solder joints.

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Current stable:
1995 E320 157K (Nancy)
1983 500SL 125K (SLoL)

Gone but not forgotten:
1986 300SDL (RIP)
1991 350SD
1991 560SEL
1990 560SEL
1986 500SEL Euro (Rusted to nothing at 47K!)

Gone and wanting to forget:
1985 524TD 167K (TotalDumpster™) [Definitely NOT a Benz]