Thanks to Joe, I'm looking at the technical manual for that system - or at least beginning to do so.
The first thing that jumps out is that the default position for the MFWD switch is to engage the 4wd. Switching between 4WD and 2WD apparently does depend on several other switches being in the proper position. I don't remember which ones, but remember reading about it, so we need to make a note of the necessary switch positions. It could be as simple as that.
The switch is spring loaded so that when the tractor is started the hydraulic cylinder is automatially withdrawn to its shortest position and the 4wd is then engaged. Apparently the normal operating condition for the 4310 is with 4wd thus engaged. Placing the dash switch in the unlocked or 2wd position should power the solenoid - which extends the hydraulic cylinder - and disengages 4wd. Yes, that seems backwards to me as well, but that's the way it is built.
The tech manual is wonderfully complete, but of course it doesn't mention anything that could be detrimental. In this case, we know that constant 4wd engagement also means that anytime a tire is replaced it is very important to replace it with the exact same OEM tire. Failure to do so will damage internal components on the 4WD drive shaft.
BTW, Which transmission do you have? I'm assuming it is not the standard shift, because on those the MFWD is manually operated. In fact, going back to manual operation by disconnecting the hydraulic cylinder from the lever would be a good test of the system. It would tell us if the 4WD system is functional.
rScotty