Another point about the round robin is that as far as I understand it, you will play more opponents than you usually do, and maybe in less time too.That mkes it difficult to get to know your opponent's team better, which is essential in a playoff series.
Only if you vote on all series at once, which is not what I would suggest. Let's stick with the set-up you have suggested: 24 teams in 6 divisions with 4 teams per division. For example, divsion A has the following teams: Penguins, Senators, Predators, Ducks, Devils and Kings. Here's how the voting would work:
I) REGULAR SEASON
Round 1:
Division A:
Penguins vs Senators (8 games)
Predators vs Ducks (8 games)
Devils vs Kings (8 games)
Division B, C and D:
Same system: 3 individual match-ups per division.
Every GM has the opportunity to argue his case against the current opponent: Penguins' GM vs Senators' GM and vice versa, Predators' GM vs Ducks' GM and vice versa, and so on.
Vote 1:
You vote on the round 1 match-ups from each division. Your ballot will look something like this:
Penguins vs Senators 4-4
Predators vs Ducks 7-1
Devils vs Kings 6-2
etc
Round 2:
Division A:
Penguins vs Ducks (8 games)
Predators vs Kings (8 games)
Devils vs Senators (8 games)
Divisions B, C and D:
Same system: 3 individual match-ups per division.
Again, every GM has the opportunity to argue his case against the current opponent.
Vote 2:
You vote on the round 2 match-ups from each division. For example:
Penguins vs Ducks 5-3
Predators vs Kings 4-4
Devils vs Senators 3-5
etc
The process or arguing and voting on the individual divisional match-ups continues until all divisional opponents have faced each other. In our scenario (4 divisions with 6 teams each) it would take 5 rounds of discussion and voting to finalize the regular season. (
Again: The main benefit is that every single participant gets to argue his case against several individual opponents. This is not the case in the current system where you're only guaranteed to get 1 vote on the regular season as a whole and then 1 vote on an individual playoff series, the latter one often being a rather hopeless case if you've been ranked low in the regular season.)
II) PLAYOFFS
The 2 top teams from each division advance to the playoffs. With 4 divisions, that means you've got 8 teams in the playoffs, all of them #1 or #2 seeds from the first stage. You will rarely get match-ups where the outcome is obvious in advance. And you will only have 3 playoff rounds overall (QF, SF, F) so that the overall length of the ATD shouldn't become too much. (5 rounds in the regular season + 3 playoff rounds = 8 overall rounds of discussion and voting.)