The biggest problem with pitted bores is that they are very difficult to clean and keep clean, rough chambers may cause extraction problems.
What you may be seeing as pitting might be just a very dirty barrel. I have cleaned up a few barrels that looked hopeless, and they turned out to be reasonably good shooters. Before spending a ton of money on reamers, you might try a couple of things. I WOULD NOT USE THESE METHODS ON A VALUABLE FIREARM. We are talking about attempting to restore a 1919 barrel to reasonable firing condition.
First use a good penetrating oil like Kroil and a new bore brush, soak for a half hour and brush in the direction of bullet travel. I have had much better luck with the Kroil than with any other bore cleaner. Kroil has a very distinct odor, but , hey, you are cleaning up the bore not going to the prom.
I have a one piece stainless cleaning rod that I use for only this purpose. After about 50 strokes (with the brush) run patches until they come up clean. Pull off a small tuft of 000 steel wool and thread it thru the patch holder so it will be a snug fit in the bore, soak in Kroil, 50 more strokes, patches, switch back to the brush. Keep repeating and checking the bore with a good light. Remember you are CLEANING the bore not trying to remove the lands, and it's only going to get so clean. When you think it's clean (it won't be) put some "Final Buff" a PPG product that is used in the automotive body shop business, or some jewelers rouge on a patch and take about 10 strokes on the cleaning rod, back to the Kroil, brush, steel wool, patches. Sometimes an overnite soak with Kroil and repeating the process will be necessary The chamber can be polished the same way using a pistol cleaning rod with the handle removed chucked in a drill. Either discard any of the cleaning tools or keep them seperate from your other cleaning stuff when you finish.
Another thing to keep in mind is a 1919 is not a benchrest rifle, what most of us are looking for is functionality, not 1/4 MOA accuracy.