I'm not a fan of heavy cleaning of my rimfires.
No brushes, no solvents unless accuracy drops off.
At the end of a morning, a couple of dry patches down the bore
is all that's necessary to remove the gsr left in the bore.
The wax lube, from my ammo, remains in the bore to protect the metal.
You'll probably get the same effect from the no-brush snake.
Just remember to clean that snake regularly, to remove the glass particles
and grit that collect in the fibers during use. Cleaning is ok, abrading isn't.
As to cleaning a boresnake, I don't use the washing machine.
No desire to spread glass particles around the tub.
I use a large mouth clear 16 ounce plastic container with a screw top lid.
Place snake in container, half fill with warm water, add 2 drops of liquid dish soap
screw on the lid and shake well for a minute or two. When the water gets visibly dirty,
empty and repeat. When the water stays clean, drain, refill with warm water
no soap, shake to remove soap from snake, drain, repeat until no more suds form.
To dry, after cleaning, I hang the damp boresnake from a plastic hook in the garage.