There's a good chance the capacitor is kaput, it doesn't look too healthy but appearances can be deceptive. Have you tried soldering it back onto the lug on the volume pot to see if it works. Essentially the way it is wired is the way it's supposed to be, so if you solder the cap back onto the lug and the tone control doesn't work then either the capacitor or the tone pot are faulty, my money would be on the cap.
You say that the volume pot is scratchy, to be expected on something of this vintage. You could try cleaning the pot by spraying some cleaner into the internal of the pot through the gap at the lugs and then rotate the pot a few times to clean out any muck. This spray works well for this purpose,
https://www.guitar.co.uk/kontakt-super-10-switch-and-contact-cleaning-lubricantIf it's still scratchy then the pot will need to be replaced, chances are the tone pot will be in a similar condition so worth cleaning it. Again if that proves to be scratchy then it'll need to be replaced.
If the cap is a dud, then that too will need replaced. It looks like an PIO (paper in oil) capacitor which are hard to come by these days but they are available, just a bit pricey. There are also a lot of PIO fakes on the market, so you have to be careful. Alternatively a different type of cap will work just as well, orange drop caps are very popular for guitar tone controls and probably just as good as PIO.
Here is an article with a bit info on caps,
https://www.seymourduncan.com/blog/latest-updates/what-tone-capacitors-do-i-need-for-my-guitarHopefully your friend will be able to help you with this.