As you know your tach gets its signal from the magnetic pickup part number re519144. I have attached a link to JD parts, where you can look it up and see where it is located. It is a low likelihood the problem is here, but I would start by testing the pickup. The sensor output voltage is AC (I know the rest of the tractor is DC), not much room to work in there the tech manual says 1 VAC minimum. You will see a green wire number 325 and a black wire.
The next thing is to test the green wire this would require getting to the plug at the back of the instrument cluster. The black wire just put your ohm meter on the wire at the pickup and the other on the engine. Recommend you disconnect the battery (negative cable on the battery or where it connects to the frame) first.
Lets hope it is not the tach as you have to replace the cluster.
I included a few other links you may find of interest.
John Deere - Parts Catalog
Instrument Panel Fogs Up : Fog forms inside instrument panel on some 5000 series tractors, making it hard to read gauges
Front End Noise : Gear backlash causes noise on 5000 series tractors
Clutch Slip on 5000 Series Tractors : Learn the importance of checking and maintaining proper clutch alignment