I've read through the rules, and I overall like the format, and props to @GrantOen for organizing this!
My only rule change suggestion, if it is possible, is G5 (https://www.chess.com/news/view/clubs-league-season-2-rules) which indicates that if there is a tied match, 1-1 for rapid and blitz, then the higher seed automatically advances.
I'm not a huge fan of it because if the higher-seeded team wins the first match, why should the lower-seeded team show up for the blitz section? Just not my preference, but I guess it does make sense, that the lower-seeded team has to win both segments.
My preference would be maybe combining total scores of rapid and blitz at the end. It would also give remaining players an incentive to win their games even if the rapid or blitz segment is locked up.
Just a friendly suggestion, but not a big deal. You are welcome to agree, or disagree!