You got it twisted. Falcon is starting to fall off despite a focused effort of a group of Falcon mains to revolutionize his game. His recovery just can't be compensated for.
Sheik, Peach and Puff are played by three of the top 5 players in the world and are seen at every major. Peach may not get used in top 8 singles, but are routinely used in top 8 for doubles in one of the strongest team compositions.
You don't see many Ice Climbers because they're so rare, and the three big players don't really fly out to tournaments all that much, but when they do there's almost always an IC main in top 8.
The meta is absolutely dominated by Fox, but you also have to understand that the current competitive scene is ruled by six players who are almost untouchable. They play Fox, Falco, Sheik, Peach, Marth, and Puff. Pretty good variety there, and considering that they make up 2/3 of every major they all attend, it's not surprising that you're seeing the same characters over and over. Doesn't mean there aren't other viable characters though.
To your overall point, I agree, Melee is not balanced well, but I don't think Smash 4 is either, and I love both of those games.