I thought the capo di tutti i capi moniker was always more made up by the press than a real hierarchy.

Sure maybe someone had more influence wit the other families and would showing up at a Commission meeting knowing the vote but I don't think 4 bosses would had pointed and said "he's in charge"

The commission was designed as one family, one vote

That was a good laugh. Thanks for posting that. Before I began, I like to say that San Jose family never had a seat on the Commission but families on the Commission did represented them, first was Mangano (Gambino), and Luciano(Genovese) it is a bit murky, then was Profaci until the mid 1940s, Bonanno represented them from then till the Colombo represented them through John Misuaca, then Chicago.

Here is what I have.
Charles Luciano 1931-1936 (imprisoned)
Joe Profaci 1936-1941 (Stepped down, or more likely forced to step down)
Vincent Mangano 1941-1951 (Killed)
Frank Costello 1951-1956 (Stepped down)
Joe Bonanno 1956- 1963 (Voted out)
1963-1968 Vacant there is dispute that either Carlo Gambino or Tommy Lucchese was serving, and more than one boss on wiretaps was saying that Stefano Magaddino wanted that spot, but at least from 1963 to 1966 the spot was vacant.
Carlo Gambino 1968- 1976 (Died of natural causes)
Benny Lombardo 1976- 1981 (Retired)
Paul Castellano 1981-1985 (killed)
Vincent Gigante 1986- 1997 (imprisoned)

Joe Massino tried to bring the Commission back, but had very minor success. Wiretaps and informants from 2012 and onwards say that the Commission is back. There may be a chairman or it may just be selected families working together but you can bet that New York families have seats.