Squeezing the plastic extrusion in your filter box with vise grips is likely the only way that you're going to stop the internal nut from spinning.
Wow, talk about the right tool for the job! I sucked it up and bought some new vise grips (but don't worry about my old ones. I found them not five minutes
after opening the new ones, as I was looking for something else. Of course!). Anyway, for the new ones, I chose the super narrow "long reach" ones. I was busy working on something else (engine guard installation, which I'll post about if I can ever edit that post down to something reasonable), so I didn't bother with the air filter issue until today.
But anyway, wow! The vise grip solution was perfect. It came out effortlessly.
Then after you succeed in removing the screw, you still are faced with the problem of any future normal use of that screw. Maybe superglue or some form of thread lock around the thread insert would keep it from spinning.
Actually that part wasn't so bad. It turns out the screws were fine, and so were the nuts. Because the end of the extrusions were ripped open a little by the time all was said and done, I was able to see what's going on in there. The nuts are acorn nuts (the ones with a dome). The extrusion is in two parts. There's the part that you can see come to an end if you look on the back of the case. That ends with a little concave shape, which receives the dome of the acorn nut. Then the nut just sits there inside the other little piece of the extrusion, which seems to be glued (or something) to the other piece.
So, rather than being stripped, I think that little rubber (or whatever... we'll call it rubber) area that holds the acorn nut eventually just lost its grip on the nut. If that happens, then it can spin freely inside there.
Once I had everything disassembled, I tested the screws in each acorn nut, and the threading was fine, as long as the nut could hold still. One of them seemed to still be securing the nut fairly well, and the other one one's end piece was almost entirely severed off. So I first tried Gorilla Glue, but then eventually just used electrical tape, because there's not much load on this screw. I taped around the seam a bit, and the top part seemed to be holding onto the nut fairly well.
So I put everything together and now I'm fully eared
I know that's a disappointment to many of you. but I promise I'll do the ear shave one of these days. I just honestly didn't want to spend the time or money right now. Especially because I need a front tire pretty badly. So I ordered it and I'll have it by Wednesday, and I'll give changing my front tire a second try (long story on the first try).
Thanks again for the help on this, and sorry it took so long to resolve.