I never scrub the threads and especially the crown.

I use the appropriate chemical and let it do it's work. If your barrel is stainless then you can leave the crown/muzzle to soak without fear of damaging the finish of the barrel.

Get a small container, like egg cup small. Place enough chemical in the container to allow it to cover the corwn/muzzle/threads once they are submerged into the container. This requires you to place the container on the ground and place the rifle, barrel down, into the container. Leave it to sit for a little while.

After some time remove the rifle and use a nylon brush to agitate the carbon/fouling. Use cotton swabs to wipe away the excess and repeat if necessary.

Try avoid any scrubbing and let the chemical do it's work. I use KG1 and it cleans 99.9% of the fouling from the crown of my rifles.

An alternative to standing the rifle into a container of chemical is to soak a cotton patch/cloth in the chemical and dab it around the threads and muzzle. The issue with this is you need to keep dabbing the muzzle so as not to let the previous application dry out.

A small tip for the threads. Buy and use a good quality copper grease. It prevent seizing of the mod and keeps the threads clean as any fouling will rest on the grease so when you wipe away the grease the fouling, or most of it, comes away with the grease.