Try to remember that one of the fans are suppose to run constantly(engine fan w/wo clutch engaged. That is not your problem.
The problem is the "OVERHEATING". Yes, a jumped wire from the fan switch to the temperature sensor would cause the auxiliary fan to come on when the car is started which shouldn't be happening. Doing that covers up another issue.
I'm trying to make sure that I understand the situation. 1) Car was overheating
2) Someone bridged/jumped fan switch to temp sensor.
3) You R+R'd fan clutch and temp sensor
4) Now your auxiliary fan runs constantly, but didn't before.
You still haven't told me the order of events. Did someone try to solve your overheating problem for you by bridging the fan? If it has always been wired like this your auxiliary fan should have always been running constantly. The connection point of the bridge may have come loose a long time ago. When you replaced the switch and the jumper it just started working as it should with the wiring. As I stated before the only way to figure out the true OVERHEATING problem is to wire it correctly. With the fan running like that it causes a constant strain on your alternator.

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1995 E420 152k
1991 300D 2.5 348116 daily driver
1992 300E 2.6 196k-work in progress(head gasket)
2002 BMW 745Li 93k