I don't have any formal proof yet, but I'm fairly confident that you can't find all players with all backs. I believe that the Kotton "Never Go Out" big scroll (KNGO) and the Virginia Extra were printed first on the thin paper. At the end of those runs, they started printing the KNIT, K123, and Mino backs.
If you look at the team changes as well as the players with only one team, you'll see what I mean.
For example, Fred Jacklitsch -- he played with Philly through the 1910 season, then in the minors for 1911 through 1913, and then went to the Baltimore Feds for the 1914-1915 seasons. He has with Phila. Nat. and Baltimore Feds cards in the set, but you'll only find the KNGO and VE backs with Phila. Nat. You won't find those two backs with Baltimore, but you will find K123 and Mino backs with Baltimore
Another example is Ray Demmitt. He only has Chicago Am. cards, and he played for them in 1914-1915. I'd love to see someone show a KNGO or VE of Demmitt, but I don't think they exist. The K123 and Mino definitely exist.
Eddie Plank is only pictured with St. Louis Feds, and he played with them in 1915. You won't find a KNGO or VE of him, but the K123 and Mino exist.
On the flip side, Lajoie Cleveland Amer. (fielding) I believe is only ever found with a KNGO back (although I think I VE could exist). Lajoie was with Cleveland through the 1914 season before going to the Philadelphia Athletics for 1915-1916. The Lajoie Portrait is only found with Phila. Americans team caption, and isn't found with the KNGO back, but definitely has the K123 and Mino backs. (I think the KNIT back could exist, but I haven't confirmed it...only my list of things to do.)
Chief Bender (striped cap, Baltimore Feds) -- he played with them in 1915, and that can be found with the KNIT, K123, and Mino backs. The striped cap with Phila. Am. is found with the KNGO and VE backs. (I'd be interested to see Phila. Am / striped cap with any of the KNIT, K123, or Mino backs -- I haven't seen them, but since he was with them through 1914, perhaps they exist.)
(If anyone has any of the backs that I claim don't exist, please do show! I'd love to be wrong here.
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Cheers,
Steve