I understand enjoying the aesthetics of traditional wood rod while loading and shooting.
When something goes wrong,its OK to use the best tool available.
IMO,it can be modern.
Stuck ball? Two kinds.
One,trying to take too many shots between swabbing the bore.An aggravating factor is using too much powder.
Extra powder is sort of like an engine running rich and carboning up.If things get crunchy after 5 rounds,back off your charge.
If you moisten your patches with something like Black Solve and have the right charge,you should get plenty of shots off without needing to swab.
Then,have a feel for it,and run a wet patch as needed.
The other problem would be forgetting to put the powder in.
How often does that happen?
I don't recall if I ever had to do this,its been decades...but seems to me by pulling the nipple out of my Griffin hook breech,or the drum screw out of a conventional side hammer, enough 3 F or 4F can be fed in behind the ball to bloop it out of the barrel.Others may have more to say about it.
I never had to use a ball puller.
Now,day in ,day out loading,etc,I would not use a sectioned rod.
But,right now,on e-bay,are sectional steel 50 BMG tee-handled cleaning rods.They should be able to pull anything.
Cleaning?I used to pop the bbl out of my Hawken.
I modified a Water-Pik hand held shower handle by cutting it off and epoxying a 3 ft piece of 3/8 brass tube in it.
I'd screw that on the hose,insert it clear to the breech.I had a tag of wire to hook to a ramrod thimble.A prep wet patch with black solve won't hurt
Then I'd turn on the hot water.
Full pressure hot water blasted into the breech ,pumping the crud out
And the barrel would be hot when I was done.
A few dry patches and an oily one,I was done.