I have some good news to report!  On the weekend I was able to get out to some private property and test the 597 and actually had repeated full-magazine dumps without issue.  It's taken a number of iterations to get to this point.
Failure to Extract - stock extractor had a bad bur and edges were not crisp.  Initially I used a fine diamond stone to remove the burs and define the edges.  This made a world of difference. It probably would have been fine but I replaced the extractor with Volquartsen part for good measure and extraction is never the issue now.
Failure to Fire - this was nowhere near as common as the others, but I polished the rods, sanded and polished the bolt surface that touches the hammer, and used dry graphite on the sliding parts before assembly.
Failure to Feed - I thought this was the main issue when first looking at the rifle. I tried both Promag 22 and factory gen 4 10rd magazine.  Then a friend gave me a custom plastic magazine.  I bought the M-Carbo magazine spring and that did seem to help reduce the cartridges from stopping in the magazine. Polishing the inside of the metal case slightly and adding dry lubricant also made a significant difference.  There certainly does seem to be a difference in function depending on the orientation of the spring.  The 'higher pressure' side of the spring needs to face forward, toward the barrel.  The action also needed to cycle faster to not give fresh rounds time to stand on end in the magazine.  Using 1235-1640 fps ammo certainly seemed to improve this issue.
Failure to Eject - Related to above, the ejection was kind of pathetic looking back now. Using 1235-1640 fps rounds of various models and manufacture, combined with polished and lubed rods cycled fast enough to really start throwing the cases with zest.  The final piece was actually backing off the setscrews behind the rods completely.  I had a hint of tension before, but backed them off at least 1/8 of a turn from touching which felt completely wrong, but it worked.
I was able to shoot more rounds effortlessly this weekend than I have the rest of the time owning this thing combined.  Many times I took the other 22LR but NOT the 597 because it caused me nothing but frustration.  In fact, I tried it one evening last week when I was out mainly to test some hand loads in a centerfire and thought I would give it one more chance will almost everything above done.  It wouldn't feed a single round.  It wouldn't even single load from the magazine.  I shot it a few times by manually starting the cartridge into the chamber.  I was about to literally throw it out, as I couldn't even imagine selling it to another poor soul like that.  5 minutes before heading out on the weekend trip I decided to try one more thing so I pulled the action and back off the setscrews ever so slightly and pray for the best... it worked.  I don't believe for a moment the setscrew pressure on the rods was the only culprit, but without that and all the other issues sorted together there was little hope of success.
Now I'm so glad that we gave it one more shot, as it's a very fun rifle to shoot (even as fast as I can pull the trigger if I like).  That actually makes it even more disappointing that I'm not able to find extra magazines now.  BTW, with all the improvements on the action, the ProMag 22 still doesn't work at all.  Now the 10rd works flawlessly and the 22rd may get cut up to harvest the spring.
I was even tempted to try making a new magazine by machining two halves from metal but reproducing the little internal feed ramps/lips seemed really fussy and awkward with the manual equipment I have (and lack of skills). So I'm still looking for magazines that are factory original, reproduction or aftermarket so long as they actually work.