So where did we go horribly wrong? Who's better than who we have ranked above them and why?
You forgot to factor in team "weight" I think. That is, teams can be broken down into either 2 1 vs 1s, an actual 2 vs 2, or the superior strategy of 2 vs 1 vs 1. That is, 2 players on a team make 1 player helpless to help their teammate, then simply 2 vs 1 that teammate since 2 vs 1 is, or at least should be, impossible.
Example, you have Zelda way too high. I play Zelda and I have an attractive win record simply because my opponent usually do not exploit her slow speed. That is, should my opponents choose to 2 vs 1 my teammate, Zelda has very few practical tools to stop them. In the words of my longest and best teammate Mew2King, "Don't let them use teamwork!".
That said, Fox should be in his own tier for teams since Fox is the only character capable of instantly changing the momentum of a match by being nearly immune to the outlined strategy above. You should also have Peach higher, since she's the most effective character against Fox in a format where any one player cannot control a match in a complete way. Yes, I think Peach should be higher solely for being effective against Fox. In this way, I think Fox, Jigglypuff, and Peach are the best characters for teams. Every team should have a Fox. For similar reasons, you have Ice Climbers way too low, and Sheik slightly too high. Sheik is well known for having poor approaches, and is particularly helpless to save a teammate despite her amazing ability otherwise.
The opposite holds true in that a character can weigh the team down. For example, I think Young Link is abysmal garbage in teams for all of the good attributes listed above that he lacks, but also that the opposing team should have Fox. It's okay to have a "favorable" match against Peach should the match devolve to a 1 vs 1 (lol) but I think it should be fairly obvious that his general lack of potency elsewhere still makes him dead weight and leaves his teammate subject to an impossible 2 vs 1 from the opposing team.