No you're alright; the KD has nothing to do with engaging gears.
Since it goes in and out of park, The linkage as A-OK.
That just leaves 4 or 5 things;
1)hydraulic pressure, or rather lack of it.
So; it could be the pump, the hub that drives the pump, or the flexplate that drives the hub, or it could be the crank not driving the flexplate, or the oil could be spewing out of an internal leak.
2)but the most likely, thing, is lack of oil
Oil level must be checked with the oil at operating temp on flat level ground and in NEUTRAL. For you, not being able to warm it up, the level should be a little low on the stick, still in N.
3) If the L/R band is broken or otherwise not functioning, you will have zero locomotion, even tho you might have full line pressure. That band is used in both Low, and Reverse,hence the L/R designation. And unless you have a full manual valve body, the tranny will not command an upshift until the governor says to, which of course it cannot do with the vehicle at a standstill.This is the second most likely thing.
You can check the pump pressure with a hydraulic gauge of about 300psi, plumbed into the L/R servo housing. The high pressure relief in the VB is set to about 275 psi. There is very little leakage in the reverse circuit, so at 2000rpm you should be seeing around 230 to 260 psi. A good pump will have no trouble making that.
4) Of course, I have heard of debris in the cooler lines preventing locomotion too.
5) the over-running clutch is MIA
These last two are least likely.
Going back to #3, that's not exactly right. With a non-functioning L/R band, you will get no locomotion in Reverse nor in Manual Low.
BUT,
The vehicle should still move forward in Drive.
If it does not, then a full hydraulic pressure test is in order cuz the tranny may have multiple problems,(assuming the oil-level is up, and the pump is spinning).