Make it 16 teams. First round would be a best-of-3 knockout.
Next round, make it two groups of 4. Everyone plays each other in their group (So, 3 games). Top 3 teams from each group move on.
In the next round, the group toppers in the previous round get a direct berth into the semis, while the teams that placed 2nd and 3rd play each other for spots in the semis. After this, we just have the semis and the final.
This format, would, IMO, solve many problems.
-Gives minnows a taste of the world stage against the top teams, but it doesnt give them so many games that the tournament gets stale.
-No 'unimportant' matches. Every match means something. Even if you have already placed in top three in your group, you have to play to try and get the 1 spot so you can go straight to the semis.
-Not too many games, not too little. The shortest path to the cup would be 7 games, the longest would be 9. This ensures that it isn't too easy, but not too fatiguing.