RCX 2.0 Firmware downloaded successfully but "Run" does nothing
Could be the button indeed. These buttons are conductive rubber pressing on the circuit boards integrated contacts. Might need some cleaning there... 

Disassembly of the RCX is not easy but can be done. You have to remove the screws in the battery compartment, then pull slightly to disengage and remove the black IR front. Once this is done, you can pull out the PCB and battery case from the top cover. Important note: the PCB is connected to the battery case through metallic blades that can easily bend and no longer make contact. So when you pull the PCB+case, you must pull on the PCB itself, not on the battery case.

Once done, you can clean the button area with alcohol and/or swap buttons to hopefully repair the Run button ("View" button is useful, but not mandatory...).