It's only a matter of time before the NFL has more playoff teams. I'd say it's getting like the NBA and NHL, but it's not. In the NFL, the short schedule means a single crazy bounce or a bad day or an inspired day from a single player can make a big difference in the standings. In the NBA and NHL, the teams play 82 regular-season games each (a total of 1312 for the NHL and 1230 for the NBA) to eliminate fewer than half the teams. And then the NBA and NHL playoffs last over two months.
I'm wondering if any teams would get byes. Since eight is a power of two, it's easy to have all the teams playing on the first weekend of the playoffs, and then it's easy to have the bracket keep getting smaller by a factor of two until there's a single conference champion. The first weekend reduces the number of teams still playing in the conference to four, then the second reduces it to two, and then the third produces a conference champion. Having no playoff byes reduces the advantage of playing in an easy division and the disadvantage of playing in a hard division.
I don't think they can just have the wildcard teams play each other while all the division winners get byes, because after the wildcard-teams-only games, there'd be six teams left in each conference, and that doesn't work for a bracket.
If they were to go back to having just two byes in each conference, then two of the division winners and all four wildcard teams in a conference would play the first weekend, leaving four survivors plus the two that got byes, getting us back to six teams, which doesn't work for a bracket.
With eight (an even number) of playoff teams, I can't think of an easy way to have an odd number of byes.
It seems to me there must be a solution for eight playoff teams other than having no playoff byes, but I haven't thought of one so far.