I'll start off by saying no, your not over thinking it at all.

I dove off down this rabbit hole already and have done exactly what you call your "more crazy idea"
I am using the build sheet link you mention above. As I stated earlier, I only have your couple hundred VINs you list there to verify against for STA trucks, but would with near certainty guarantee the VINs I provided to be correct. For the SHAP factory I have confirmed this same process against over 14,000 VINs so far. I am doing pretty much exactly as you describe above and checking every possible permutation of the sequence. I then analyze those returns. This only has to be done once for each sequence number. Then I have the complete VIN and can just check against the correct VIN from then on.
I'm not sure if I made it clear in my first post, but I only started with the sequence number for all 235 VINs in your list (count from yesterday). My approach gave me matching results to the 209 FULL VINs you already had so that is 209 confirmations that it worked correctly.
Another thought, there is no downside in trying the VIN's I provided, worst case you still get no build sheets for the trucks, which is what you are getting now with only the sequence anyway. Hopefully we will see some of those trucks move down the line soon and it will help confirm this approach works for STA like it does for SHAP.
As for your issue with the links not working on occasion, its a cache problem. If you clear your cache before each pull you will get fresh results. Since the download is a PDF and not an actual webpage, the "VIN NOT FOUND" PDF for each VIN will be stored in your browsers cache and returned the following day when you run your scan again even if there is an update version online.