All you need is to not have any "unwinnable" matchups in order to be usable without a secondary; you could lose every match 40:60 and not need a secondary (that character would also be bottom tier, but it wouldn't need a secondary).
Meta Knight and Snake are non-controversial. Mr. Game & Watch definitely works; there's nothing so bad that he's unable to overcome it. I don't think anyone really totally shuts down Marth, Falco, R.O.B., Olimar, or Diddy Kong either. I can't imagine Toon Link actually having an unwinnable matchup. The Pokemon Trainer should be able to be smart with switching to avoid having any unwinnable matchups (I know some characters like Mr. Game & Watch have the advantage on all three Pokemon, but I don't think anyone completely destroys more than one Pokemon). I'm not even sure that most of the cast needs secondaries; secondaries are helpful for everyone of course, but I just don't buy that most characters actually need them. I mean, obviously Donkey Kong, Bowser, and Fox need them pretty badly, and bottom tier guys like our buddy Captain Falcon might "need" one in the sense that actually fighting random top tiers with him is about the equivalent of being hard countered (though it's more a statement that he's not really viable at all so any plan that involves using him less is good for the player).
Honestly, while having the upper hand in any given matchup is always good and you won't always have the upper hand if you stick with one character, it's really a very plausible plan for the mains of most characters to just focus on playing better to overcome any bad matchups they have. If you happen to really be into someone like DK, sorry, but DK isn't typical. I think a lot of people decide certain matchups "need a secondary" as an easy way out so they can just give up and not bother improving; it's really unfortunate. As a Mr. Game & Watch main, I've run into talented players who use guys like Sonic and Jigglypuff who I am supposed to completely destroy, and indeed I can see I have a big advantage in those matchups. However, I have also lost games to those characters, and I've been really threatened in general by them. Contrary to popular belief, while my tools are effective against theirs, they still do have tools that they can use to cause me trouble. Those are definitely matchups that can be overcome, and if those can be overcome, I'm skeptical that very many matchups at all can't be.