Ok...so in receiving a gruppo from overseas, I found that the cable adjuster bolt is not what I would like it to be...questions:
1. Does anyone have a suggested repair
2. If replacing, how do you take that rivet-y thing off and install a new one.

Any advice would be greatly appreciated.

The pressed-in nut is likely knurled and pressed tightly in place, so might not retain itself later if you were to press it out.
Look at the direction the cable tension applies to it, once it comes out it will probably not hold so well later!

The adjuster itself almost looks like the low-end pot-metal version, installed as a replacement. Verify by removing it and inspecting the threads for a casting seam. These are weak and unsightly as compared to the original part.
You can polish pot metal but it does not sustain a nice finish for long. The good ones are prized little pieces.

Possibly your adjuster is the good chromed steel one but has lost it's chrome?

Problem is that it simply spins and doesn't catch threads at all. I can't remove it either...?

The way i would remove it would be to saw it off. probably using a small hacksaw blade or jewelry blade. Look online for options unless you're in a big city. Can even wrap the blade end in tape to make a handle if you don't want to buy a holder. Maybe original threads are still ok... But if needed, re-tap the hole for proper adjusting screw, (which might require adding material depending on what was done to put existing piece in place.) Sorry for your discovery, amazing what people will use to "make do", and neglect to inform next buyer about... I try not to inspect bikes at night for this very reason.

I would attempt to grab the adjuster with a Vice-Grip, using some inner tube to protect the part and to provide grip.

Then I would maintain high pulling force while turning the adjuster, hopefully with some of the threads engaging below where the (cast zinc?) threads are probably stripped off of the adjuster.

If this works, possibly some limited amount of adjuster travel can be exploited engaging a few remaining threads on the adjuster

The adjuster appears to have been screwed all the way in with the soft threads having sheared where they engage the nut, but seems there is some threading below the nut threads still holding the adjuster in.

You are a genius and a gentleman sir. I was able to back it up by WD-40 and a pair of grips holding the nut portion while turning the adjuster. Might actually be able to still use it. Thank you, sir!

Nice of you to follow up, jdawg!

Hopefully the parts are serviceable. You were wise to put some penetrant in there to help get the parts apart without damage.

You were absolutely correct that someone tried to keep screwing it in and it stripped those parts of the threads. Might be able to use it if I start it higher when I install the cables. Otherwise, might have to pirate for a new adjuster. The pulleys are closing in on replacement so maybe a parts only find on ebay might do the trick.