I have used oxalic acid to remove rust stains from stonework; never on a prepainted (?) metal roof. Since the alternative is repainting the whole thing, maybe it's worth a try?
Casey

You'll want to address the cause of the staining as well. If it's due to corrosion from dissimilar metals, it will lead to bigger complications than astetics. You might seek advise from a rpeutable metal works or metal roofing outfit in your area. It would be worth a consultation fee for a pro to get up there and take a close look and inspect for corrosion issues.

You need to determine what kind of coated metal and what kind of paint. To me it looks like a factory-applied paint on galvanized steel or Galvalume rather than tin on steel.

It is important to determine if the staining is from incompatible metal chimney flashing, creosote from a chimney leak (upside-down metal chimney connection) or the rusting of exposed steel or all of the above.

The cap and pipe look much newer than the storm collar and flashing. The underside of the cap doesn't look like it's even seen a fire yet, so some things have recently been replaced- was that by you or your seller?

This type of damage happens because of bad burning habits- burning unseasoned wood or not burning hot enough. It looks like excessive gooey creosote mixed with rain water turns into something that eats away galvanized metal and rusts it out.

Regardless of how you clean and or repair this unless you change the way this is used it will happen again.

I will link below to the cleaning instructions that were given to be for my new metal Galvalume roof.
Whether or not it will clean this sort of mess is another matter- I would have a metal roof repair guy take a look at it.

Here is a link that might be useful: Painted metal roof care and maintenance