In this game my working theory is that viability of a character at a very broad, base level (before we get into the specifics of their more detailed interactions [i.e. their interactions on a vs. character basis]) is contingent on their ability to do three things:
1) bid for stage (this shouldn't be a surprise; if you can't bid for more stage in an effective manner you'll lose ground vs the characters who control it more reliably / constantly and then you'll lose because not having stage is how we die in this game).
2) maintain advantage -- either directly or implicitly; obviously the former is better but beggars can't be choosers. An example of the former would be someone chain grabbing someone or a random Falco going full Falco-master on someone. An example of the latter would be some of the traps HBox does with his air mobility and Fox in general.
3) how well they can navigate back to fair / advantageous positions (i.e. how bad is it when you're losing ground / above someone / offstage with <character>). This one is a bit wonky because some characters like Samus & Peach don't incur heavy punishments very routinely and therefore have numerous chances in weak / losing positions to return to fair / advantageous positions. Other characters like Fox & Sheik don't actually have many outs when they're in a losing position but can make use of the minor gaps in a combo or hit sequence pretty effectively, which gives them a specific thing to look for (or try to set up) when they're at disadvantage.
To his credit, Ganondorf can do 1/3; I actually think his punishment is passable. This makes him better than the characters who can do 0/3, like Roy and Zelda. That said, 1/3 is not a passing grade. In fact, at this point in the metagame I am not sure 2/3 is a passing grade either.
If you look at the characters that have gone up and stayed relatively stable (like Pikachu, Peach, and Yoshi, to cite more recent cases, and ICs for those who were around when Chu was first figuring out the character) these are all characters who can, in some way, bid for stage control (usually on the basis of speed but when Peach evolved further hers was more about stage control & using turnips better), have obviously strong methods of maintaining advantage (Pika's juggles [that can go offstage into gimps cuz of his recovery], ICs' extended grab links & juggles, Yoshi's uair spam & egg abuse, and Armada [like, in genera]), and are either flexible enough to avoid terrible positions (aMSa's platform style for example is a good way to avoid being CGed by the characters that beat Yoshi by CGing him) or don't lose badly enough in their losing positions for being put in that spot to invalidate the character completely (Pikachu's recovery for example lets him survive for a while even after he's been grab-comboed or Falco'd to 60%).