Mo,
Jointed rods can wear throats & crowns with extended use, not recommended.
But....
Let's just try this.
Run a couple dry patches through the bore with your STAINLESS rod, to get out whatever particles might be there.
Then run a couple lightly oiled patches through.
Then one more dry patch.
See if you're still getting particles.
If you're not, I'd say your particle problem is your coated rod.
If you DO still get particles appearing, you might try T's method.
I like to use the simplest approach to a given problem first & then escalate if necessary.
The only time I've ever gotten shavings, which is what your particles sound like, has been with soft aluminum rods, and they came out (as shavings tend to do) in small spirals.
On occasion, I've removed severe lead deposition in strips & flakes, but those are quite obvious.
In all the military surplus rifles I've gone through over the years, including several Mosins, I've never encountered spiral particles like you're describing.
They don't sound like carbon or corrosion.
If they're semi-hard, spiral, and of the same color as the coating on your rod, my guess is your rod as the cause.
If you show corresponding wear/scrape patterns on your rod, you run the patches I suggested with a stainless rod & the particles disappear, I'd think the problem with particulates is solved.
You can stick with your coated rod & not worry about it, or switch to a one-piece stainless rod & not worry about it.
And you can still try any of the bore cleaning methods discussed if you're getting rust out of the bore.
Denis