Wilson sympathized with the Klan as depicted in Birth of a Nation, but he was too young for the old Klan and already in the White House when the new Klan was founded.
And I really doubt Coolidge was a Klan member.
From what I’ve read it looked like Coolidge was the victim of a smear campaign. I saw several allegations but never any evidence.
Wilson was a stone racist but the evidence against him was just as slim.
I never researched McKinley.