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Could the pitting near the muzzle be addressed by counterboring just enough to clean up the pitted part?

Many years ago I had an old Iver Johnson pretty much identical to your H&R except it was a
"hammerless" DA. Had to replace the transfer bar as it was broken from much dry firing. Got it working and actually carried it some when I was working in the woods. Killed a medium-sized timber rattler with it using home-brewed shotshells. They shot pretty well in the almost-slick worn out bore.

Very cool project. I have done some rust removal of firearms and tools over the years using several methods, vinegar is one of my favorites but it all depends on what the job is.

I learned about the electrolysis process when cleaning some Winchester 1886 barrels that were picked up at a garage sale and it worked very well on them. Then a dozen years ago I picked up a pitted, rusty Police Positive Special in 32/20 that got put through the electrolysis rust removal method very successfully.

Then, last year I was given a collection of socket wrenches that had been left outside and were quite rusty. Some were wire brushed, some were soaked in vinegar and others were soaked in Evaporust. Could tell very little difference between the vinegar and Evaporust other than the latter were a bit easier to clean the residue from afterwards. Obtained the best results on these by wire brushing after soaking overnight in whichever solution was used.