It's also possible that the ignition switch itself is at fault. The starter contact doesn't have to carry a lot of current but it wears nonetheless. If the switch contacts corrode just enough, the starter solenoid won't engage. The switch can be repaired or replaced, about $100. If you get it from a dealer, it will come with a new key and it will match your existing keys. They'll need your VIN. My '96 E300 had the identical problem.

Jeremy

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"Buster" in the '95

Our all-Diesel family
1996 E300D (W210) . .338,000 miles Wife's car
2005 E320 CDI . . 113,000 miles My car
Santa Rosa population 176,762 (2022)
Total. . . . . . . . . . . . 627,762
"Oh lord won't you buy me a Mercedes Benz."
-- Janis Joplin, October 1, 1970