Czuch: It looks like things will sort themselves in the next round.... that nobody will get a bye then,

Incorrect. For the first tournament, there are 2 matches, and 4 byes. So the round will have 6 players (winners of the two matches, and the four people with byes). That means there will be 2 byes. Now, it may very well be that the two byes actually "play each other" ("winner" of 13 against 14 against "winner" of 15 against 16), but that means there's still a bye in the round after that.

It's a complete mess.