Just a thought...buy a cheap pinball harness off of an auction site like ebay. You will have all of the correct spare wire you will ever need and there is an excellent chance you will have the correct color coded wire you are replacing.

I can't stress enough how important it is to use the correct color wires for future diagnosis and repair. Unfortunately the correct multicolor wires will be next to impossible to find unless you can find a replacement backbox from the same machine model. In those cases use single color wires with colored electrical tape or even better, small pieces of heat shrink tubing for the 2nd and 3rd colors.

You don't need wire that matches all 2 or 3 colors, just a few monocolor spools to match each main wire color. Then match the 2nd (and 3rd) wire colors with small pieces of colored heat shrink tubing.

It would be a good start to identify the specific wire colors that match those originally used in your machine and also the approximate lengths you might need. Someone in the forum just might have the appropriate wire colors in their parts bins or know of a vendor that could supply something close.

You could do that by tracing backwards from the burned sections and/or verifying with the schematic.

Search YouTube for soldering videos.
You’ll want to use a decent sized iron or gun.
A wire stripper makes it easier, although with cloth covered wire you can often push the insulation back enough to give you bear wire.
Use solder with flux in it (rosin core), not plumbing pipe solder.
Remember to slide the shrink sleeve tubing on before you connect the wires, then after you’ve made the connection, slide it over and shrink it with a heat hub or heat from the soldering iron.

You’ll get the hang of it.

Dave