Don't over-think the process.
About the only part of a barrel that gets hurt in cleaning is when there is carelessness at the muzzle: Some sort of wear on the crown. The brass cone-shaped dealies through which the cleaning rod can freely move make for the best protection.
The wax on most .22 rimfire ammo pretty well protects the bore against rust. Many and many a .22 rifle gets cleaned about once every blue moon, and other than for precision target shooting, does just fine.
After whatever cleaning I do--rimfire, centerfire, whatever--I run a clean patch through as a check. I'll then spray a little gun oil on the patch and run it through as a rust preventer before storing.