A multitude of problems can crop up if you do not have a good ground from the battery! Have them check that.I've had several electronic/computer glitches with this van ever since I bought it. And sounds like your reading lines up with what the mechanic on the phone was telling me about the no start issue.
I took the van on a week long vacation and was first stranded 300 miles from home with my gearshift locked in park as well as having this no start issue. Both of which I'd experienced before but not lately. I had already learned how to get the gear shift unlocked with the unlock button on the dash and I talked on the phone with the mechanic, right before AAA showed up. I had sometimes been able to resolve the no start issued by fiddling with fuses and relays under the hood. Basically wiggling and pushing and just making sure they were connected. Sometimes that would fix the stuck in first gear issue as well. And I was usually seeing codes. 882 I think was one and another in the 500's.
Anyhow, the first time I was stuck last week, I had already gotten the gearshift unlocked by the time AAA arrived. But before loading it on his truck, the driver and I both pulled and pushed every fuse and relay in the box. And by some miracle, the dang thing started and I was on my way. Later in the week, I had pulled into an Interstate rest area and shut down my engine for a minute. Then cranked up and pulled back out onto the Interstate and the stuck in 1st gear happened. I tried all the usual tricks that had sometimes worked in the past but nothing worked this time. So this time AAA loaded me up and carted me off to a dealer. They checked it all out and said the only problem they could find was the faulty transmission relay. So I was good for a few days until the no start issue reappeared (but no codes this time).
I finally made it back home last night though and I have two other more reliable vehicles to use, so I'm giving the van to the local dealer tomorrow to keep until the issue happens, so they can figure it out. I've cranked it up several times today though without a problem.
I think you're right though. This particular glitch sounds like either the Key fob or the WIN module. It's just hard to figure out an intermittent problem without just replacing parts willy nilly until the problem goes away. And even then you're never really sure it's fixed lol!