580SK starting trouble

I have a 91 580SK, and I have had intermittent starting issues since I have had it. I was able to deduce early on that it had something to do with the neutral start switch. Well, for years now, I always have been very careful to make sure the thing was in neutral on the shuttle shift before turning the machine off. Most of the time no problem. If there was, a quick rattle back and forth on the shuttle shifter has usually remedied the problem. I recently replaced the alternator,as I was using it and starting and stopping from place to place. I noticed that it was dragging a bit when cranking. Huh... One more stop and I cranked it and it gave about a little grunt and no start. The volt meter indicated low voltage battery, so I put my charger on it for 30 minutes. Started right up. Must be the alternator as it showed no charge indication when running. Changed out alternator.
Finished the swap, and put charger on to help batteries, and no start. The usual rattling of the shuttle shift produced no result at first, then as I toyed with it the machine started to crank and it started kind of in reverse. Wow! Wel I let it run and noted the battery in charging condition, a good sign, and I ran it for a little plowing a little snow then put it away, and shut it down. Well, now it will not stert no matter waht I do. Tried remote start button etc., no good.

What did you hook the remote start button to? Put a multimeter on the battery with some clips and see what the battery voltage is sitting, and when you try to start it.

If your battery voltage is fine, then go to the starter and see if the solenoid is getting a start signal to the small wire while you try to crank it. Then go to the neutral switch and check if it's working.

I put the remote switch across the battery term to the switch term. I did this one time before and it bypassed the NSS, this time no good. The batteries are sitting at 12.5v. I pulled the floor plate out, and I believe I am looking at the neutral start switch. The shuttle linkage is right beside on the left. Funny thing about this is, I was working the shuttle shift back and forth while the key switch held on start. I can hear the cab relay operate, and as I move the shifter back and forth slowly I can hear what I think is the NSS.

If you bypassed everything at the starter then you need a solenoid or a starter.

you may want to put a volt meter on the hot side of the starter relay and when you try to start with remote start switch the voltage may drop to zero, indicating you have a bad battery cable

Checking voltage drops is a quick way of telling the integrity of your start system. Start at the battery and put the volt meter leads on the positive post and the hot lead on the starter and try to start the machine ( you can only check voltage drops with current flowing) The higher the reading (volts) the more resistance you have.Then you break it down to smaller pieces to find your high resistance.
And don't forget the ground !!!

This is strange. I repaired some of the wire loom having to do with the NSS. However, that was not the problem as continuity was good before I did that. I started checking voltages at several points. I can put the vom on one side/wire on the nss, and with the key switch on I read my voltage on the one side(red wire) but not on the other side or pin. When I turn the key to start position I lose the voltage. Now where is this taking me.

do you have the wiring diagram for the machine?
I'm not sure but the NSS may provide a ground path for a start relay (not power directly to starter solenoid, you mentioned a cab relay in earlier post and I am not sure what you are referring to) that would in turn provide power for the stater solenoid.
If you jump the starter solenoid (hot to start post) is the starter working?

Here is part of your starting motor schematic from the service manual.

Looking at the schematic the path to ground for the starter relay is through the NSS

I agree with caveman. I have found as a rule that the more intermit ant your problem the more likely it is a ground. If I had a nickel for every time a machine wiring problem turned out to be a ground my name would be donald trump.

Here is more crazy. I pulled the starter as I supected it may be in trouble as I have already replaced the altrnator back when I thought that was my original trouble. Had it tested and the starter was toast. Matter of fact it is a wonder it even did start. OK. New starter installed. Go to try it and nothing. WTF! Now, I try again and click click etc.. Again I try the shuttle shift shake and wallah, it starts. But with the shifter forwaerd just past the neutral. Huh. So I fgure I'll move the thing into the main shop to complete this fix. I ease back on the shuttle control to reverse, and ZING! it's engaging the starter. Holy crap is this thing possessed? Now I'm still back on the original of the neutral start switch. I didnt think at he time to pull the wire loose from the solenoid and drive it in the shop. I have ordered a NSS, and I'll go from there.

Might be an idea to check the transaxle mounting bolts, there are only 4 of them.
Your whole transaxle maybe moving to & fro.....