About the discussion-worthy players: I think I have an opinion on each of them but I didn't think they were exactly obvious so it was worth talking about them. Here are the ones from Iain's 1920s list (which means they are perhaps the only ones who need to be discussed, as I doubt any 1910s or 1930s-associated player would even be arguable)
Hainsworth: 1895 born,
Cook: 1895/96 born
Oliver: 1898 born
Noble: 1896 born
Gardiner: 1891 born
Dutton: 1897 born
Cooper: 1899 born
Goodness, one thing that stands out right away is that Gardiner is clearly a pre-merger player, as he was 34/35 when he joined the NHL.
It's worth checking the years of birth for some others to see if they are from the same few birth years.
Duke Keats (1895)
Buck Boucher (1895)
George Hay (1898)
and I guess we have to throw in Art Gagne (1897) too.
Everyone else seems obvious to the point it's not worth discussing. So our 1895-1899 birthdays are:
George Hainsworth (1895)
Duke Keats (1895)
Buck Boucher (1895)
Bill Cook (1895/1896)
Reg Noble (1896)
Red Dutton (1897)
Art Gagne (1897)
George Hay (1898)
Harry Oliver (1898)
Carson Cooper (1899)
Is it possible to draw a line based on year of birth? Or does it require a case-by-case look at how many years they played before and after 1926? Does that mean we're really just basing it on what age they retired at? Because they all started playing hockey that mattered right around the age of 20.
Off the top of my head, I associate Hainsworth, Cook and Cooper with post merger hockey for sure. But that includes the oldest and youngest of the ten! Keats and Boucher seem obviously pre-merger.
Hainsworth: 13 full seasons before 1926, 10 seasons after (but the ones that he is best known for were after 1926)
Keats: 9 seasons before 1926 (would've been 11 if not for WW1), 3 after (plus 5 in a noteworthy minor league), all his best known seasons were before 1926
Boucher: 11 seasons before 1926, 6 after
Cook: 4 seasons before 1926, 11 after (2 of his best 3 seasons were before merger, but 9 of his best 12 were post). We could say he had 6 seasons before 1926, if we even know what NOHA and NMHL stood for. And what the heck was he doing from the age of 18 to 24???
Noble: 11 seasons before 1926, 7 after. 1925-26 was the first year of the twilight of his career which he spent as a hard rock defensive defenseman. He was a high scoring forward prior to this.
Dutton: 6 years prior, 10 years after. He was a WCHL all-star in his 20s, but also ranked highly in Hart voting a few times in the NHL in his 30s.
Gagne: 6 before, 6 after. His pre-26 seasons are his 1st, 2nd, 6th, 7th, 8th best.
Hay: 5 before (7 if you count SSHL - we probably should), 6 after. His WCHL/WHL seasons are his 2nd, 3rd, 4th, 6th, 7th best.
Oliver: 6 before, 11 after. His seasons before include his 1st, 2nd, 4th, 5th and 7th best.
Cooper: 8 before, 6 after. He's tough because he was an absolute smokeshow in the OHA but is best known for being an NHL player (which he also was for 2 years before 1926). My numbers say his 2nd-7th best seasons were before 1926.
I don't think there's any way we can go by age! Everyone's career is different. I think Keats and Boucher are as obviously in as it gets. My initial instinct to exclude them was right. Dutton and Cook are as obviously out as it gets.
In between that, I'd subjectively rate them as follows:
PRE
Noble (61% of seasons pre-26, 100% of significant seasons pre-26)
Cooper (57%, 67%)
Gagne (50%, 50%)
Hay (45%, 63%)
Oliver (35%, 55%)
POST
From that, I'd say Noble and Cooper are in, Gagne is in because 50% should be good enough, and besides, he had four significant seasons so he should not be a threat anyway. Hay and Oliver had less than half their careers before 1926, but more of their significant years were before 1926. On the basis that at least the 2nd numbers are over 50%, I'd say keep them in.
I left out Hainsworth because he's harder to judge. I have no formula to just determine when he was truly at his best. Surely, his most famous years are 1927, 1928 and 1929 (when he won the vezina despite not really being known as one of the few best goalies), and 1930 and 1931 (when he won the cup and a retro smythe). But the years in which he actually got some individual recognition were 1918 & 1920 (OHA 1st AST), 1921 & 1923 (OHA 2nd AST), and 1926 (WHL AST). Is all that enough to overcome actual NHL achievements, as team-oriented as they were? I dunno. He was 31 before he played an NHL game.