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Joe_G.Couple comments in no particular order.
Hal, I see how you play the "highest graded" Old Judge game 
But how do you know whether the higher graded examples are either later year issues or different poses from the "rookie" year? None of the grading companies seem to track this information. For example, there are two higher graded Monte Ward cards with another two graded equal to yours. Are you sure the other four examples are all different poses/later years?
In all honesty, I'm just trying my best to give you a hard time in good spirits 
Your HOF collection is top notch. Your SGC80 John Ward is BEAUTIFUL.
John, Fred, others, thanks. John, please no nagging.
Kevin, great points. However, on point #3 I still rather report the average card per player as I did. Let me use a hypothetical example to help demonstrate why I calculated the number the way I did.
Let's say I collect 1984 Topps because of my fond Tiger memories. There are 792 cards to a complete set. I collect them by opening wax packs and quickly realize I need to open more than 792 cards to complete the set. After collecting 1584 random 1984 Topps, I still need 36 cards. Now at this point I could report my collection as any of the following:
1) I have an average of 2.0 cards per player in the set (1584/792)
2) Of the players I have, on average, I have 2.1 examples (1584/(792-36))
3) Worthless, must sell post-war and buy pre-war:)
All 3 are accurate, really just personal preference. In the case of Old Judge, you could use the numbers from the first calculation and begin looking to see which side of the average the 520 cards fall. The cards with zero & 1 populations are certainly higher probability to be the tougher cards etc. However, as stated before, the population size isn't the best for statistical analysis. In other words, the confidence level is lower than I'd like to make statements that all 0 & 1 population cards are more scarce than the cards with a 4 population (for example).
Nick, Bid McPhee is one of the tougher HOFers for reasons you stated and more. His major league career was well underway by 1887 but Goodwin & Co. only issued cards for 3 of the 8 American Association teams (NY Mets - Spotted Ties, St. Louis - Brown's Champs, & Brooklyn). Had they included Cincinnati in 1887, Bid McPhee would likely be an easier find. This doesn't mean that all HOFers issued "rookie" cards after 1887 are difficult, it just helps to be included in the more common 1887 run. The other factors you listed are valid as well.
Also, the managers and most other Old Judge subsets seem to be in demand by collectors. Since completing the set is about as close to impossible as it gets, many collectors find sanity in trying to complete a subset. Managers would be a good example of a subsets. Most managers are also portraits which fall into another subset category. The Old Judge set, more than any other I can think of, is collected by subsets.
Lastly, Mr. Maples, your SGC84 Hanlon is gorgeous!, thanks for sharing a pic with me awhile back. I want you to know that I'll be sending you some scans of my Hardy Richardson duplicate as soon as it comes back from SGC. Do you want to use that as credit towards your Hanlon?
Regards,
Joe G.