On a W126, you don't have to move the tranny to access the neutral start switch.

First thing would be to make sure it's properly adjusted and that the retaining bolts are tight. Loosen the bolts and swivel it while testing if the engine will start and that the reverse lights come on when the gear lever is in the appropriate position.

The bushings will be on the rod that connects the bottom of the shift lever with the transmission. No bushings? The old ones probably fell off.

If the rod is at all loose, fix that before touching the neutral start switch.

Sixto
91 300SE
87 300SDL
83 300SD