Wire rope OA

Not knowing much, I would use brass brazing, it should flow pretty well.

I've seen cable bonded using silver solder and there is a system that uses Zinc (I think) and a tapered socket. Spelter Sockets, I believe they are called.

The standard method works fine:

Aluminum crimp sleeves should be strong enough for a hand-pulled cable, if you have the right crimper.

I've played with most of the do-it-yourself methods to find one that won't snag when pulled over a high tree branch. This was the weakest which slipped at a little over 100 lbs:

There is an epoxy version of the spelter clamp that gets good reviews, and one with a wedge socket which I've seen used on a crane, but I couldn't find under 3/8". AFAICT properly applied U bolt clamps are as good as you can do unless you have a local shop that will install swage fittings.

If you spin that cable the wrong way (untwist it), you'll have more problem than a dirty chimney. Might want braided cable.

It is also done in the field. Both zinc spelter and resin (I assume an epoxy of some sort). If you google on Wirelock Technical Manual you can find an extremely detailed manual on the subject.

Do you really think circular brushing will work when linear motion doesn't?

How do you plan to move the spinning brush up the flue? A ladder above the chimney?

An unsheathed spun cable will knot up unless you can maintain a very strong pull on it.

Depending how well the late night fires burned my chimney can accumulate between imperceptible and half an inch of soot in the chimney per week. I monitor it during the day, with a thermocouple readout in the kitchen and a mirror outside a window to see the chimney top. Usually my home-made nylon brush cleans to bare metal but occasionally there is a hard glazed layer that needs a good scrubbing with a tight-fitting wire brush on a sectional rod.

If the ball isn't a tight fit then an eye nut plus a locknut should work OK. You could wrap it with duct tape to stiffen the connection and protect the flue from the hardware.

A "Hickey" from the hardware store.

My brush has 1/2-20 threads, or at least it does now after cleaning up all the joints with a tap and die.