Hello Thomas,
It is indeed unfortunate that your track has degraded to this state !
However 1200 grit sandpaper or even a finer grade if you can find some, would be the most likely thing for the job of cleaning up the centre contacts. !
If you search this forum, there are a few threads about it, I just did a quick search but there is too much to wade thru to find the answer I was looking for. So I went to youtube and found this from list member "Schwipple" altho we haven't heard from him in a while
However your problem isn't dirt or oil, so being rust, I would substitute the microfiber cloth shown in the video for "Crocuss Cloth" which is an old abrasive cloth I remember my Father having, but I am not sure it is available anymore, these days ! So my suggestion is to glue or fasten somehow, the 1200 grit or finer sandpaper to the pick up shoe, or slider, of a small wagon as in the video, and by hand slowly run it over your tracks, maybe going back and forward a little bit over small sections of the track at a time ! Even put some of that Labelles 108 oil on the sandpaper if you have some
Rubbing the sandpaper on the studs with your fingers is too imprecise and your likely to be rubbing the rails as well which is a big no - no
Do not over do it as you will take too much material off the pukos !!.
This will leave your studs clean but liable to rust again at the slightest onset of moisture in the air, even heavy breathing with the excitement of your trains running on the layout will be enough
So to counteract this I would apply a slight smear of petroleum jelly (Vaseline) to all the sliders of your loks and wagens and run them around the layout and that should leave a small enough coating on the studs to prevent the rusting again, and keep reapplying the petroleum jelly from time to time, but I am talking about just the finest smear with your finger !
I used to do this with my childhood M track layout to reduce wear on the sliders and never had any problem after long periods of disuse

I also intend to use this on my new K Track layout I am building !!
If too much jelly builds up on the studs, but it shouldn't

Go back to the video and repeat those steps exactly !
And just btw, I use a smear of Labelles 108 oil (as shown in the video) on all my sliders of my Loks after running on my current C Track "test layout" and before returning Loks to their boxes, I should probably apply it before as well, but it is recommended as a rail cleaner ! but as I said, I am intending to use Vaseline on the sliders for running on my K Track layout. Only a very fine smear is required, just the slightest trace is all it needs to protect the metal of the pukos or studs !
Best of luck with getting your layout running