Your spare black wire is exactly that . A spare . It is intended for use in
setting timing .
As for setting timing with new points . The fibre heel on your old points will
be worn down some . Just setting in the same position with new points , won't
give you the same setting . With RT's the timing is critical . Advanced timing
increases the chance of kickback when starting , sometimes with serious results ,
for either the rider or bike . Setting with a dial gauge is best .

When I cleaned the points on my CT-1, I just put a pencil mark on the stator plate where the notch was, took them off and cleaned them, and put them back on where they were. It's been running great, but Swoop has a good point about the consequences of getting it wrong on a 360.

Ok, Swoop56 is correct of course. But in the interest of keeping it simple & seeing as the points gap is your only timing adjustment in your case as the backplate is fixed----set the points opening going by the marker by lining it up with the flywheel notch. Have them JUST starting to lift right there. Could then check max opening by turning it a bit to the high point but should be within 12-15. Tighten that flywheel plenty tight, or bad things happen. 'Just opening' is 'just' that. If ya can see daylight its too much. Cigarette paper thickness. Backtracking==if the timing plate that hangs off the lighting coil is obviously bent by someone playing then ok ignore my woffle. Think it originally sits pretty much square on to the screw. Could check the opening point by using that wire your holding & plug in a multimeter [old style analogue].
Getting back to the old points though--points don't go bad, well not unfixable bad, unless the buttons drop off or the heel drops off. What makes them bad in your case.
Sorry--lot of writing.
Writing style changes after a few lines, no idea why it does that. Annoying. Not hitting anything i'm aware of.

I'd find a good two stroke guy at a old beat up shop and have him change the points and set them, plus the usual carb cleaning/rebuild. It's best to have a mechanic set up a RT if you want a bike that starts "first kick.
Nothing to play with as the kick back is tough. Use the heel of your boot when you kick it not the boot arch, if it kicks back it hurts less lol. Nothing sounds like an RT at idle.

With new points, if you set them & the timing at 2.9btdc they'll settle in & gap will close a whisker & timing might come back to about 2.5 which might be a good thing, good for kickback.

As others have said, if bike was running good simply mark the flywheel and case (or use stock timing pointer) to show where points are opening now.  Then set new points to open at exact spot. 

But why change them?  More problems happen changing points.  Unless they are somehow destroyed just a cleaning should restore spark.  Don't even need to remove flywheel.

Maybe I'm just a cheap bastard but I don't change points unless they really need it.  Especially these days with some of the junk replica ignition parts out there. 

As Swoop says, Simon, leave that black well alone. It is not an earth and you will lose spark if you treat it as such.

You only need to use the white wire if you're suffering from a low charging rate, otherwise just leave it.