Ohhh, gosh, looking at that disk I was feeling terrified. I have never bought a disk in that state :O
Take care with the places where they polish disks. I don't know if it's expensive over there, in Spain it's kind of cheap, like 3€, but it can ruin the disk completely. It will literally make it thinner by removing part of the surface, and if the machine isn't good or the worker does not know how to operate it correctly, say bye bye to the disk.
The toothpaste trick is nice if you have disks with just few light scratches, I did that to dozens of second hand games bought here and their look defenitelly improved. From time to time, if I buy a game with light scratches I do that. Well, now I have a more professional polish liquid, but it's just a little bit better than toothpaste I tell you.
The way I do it is: I take the disk and I put 4 drops in every side of the disk, centered. I take a make up cotton like these:
They usually have 2 faces, one which is softer like cotton, the other which is more like a soft cloth. First, take the cotton side and go in circles around the disk, always in the same direction. Do a 360º round like 3 or 4 times, until all the toothpaste is evenly applied.
After that, take the other side, and do the same exercise, until you see the toothpaste is going away little by little, until the disk gets a more "shiny" look. If you still see toothpaste, do another round, until it looks like nothing has been applied. Don't press too much the cotton, or you will remove the toothpaste completely, do it gently all the time.
After that, try if it is read by the console. If it's not or it takes a lot to read it, do another round, very gently, until you get the desired result.
That's my way and I never had any problem, but it's true I never applied it to a disk so damaged :(.
I wish you the best luck!
NOTE: Make sure the disk is very clean. Do a clean up with some isopropyl alcohol first.