The answer is technically yes, Yoshi can definitely be solo viable. Even outside of Zero's video, and into this board, we agree that Yoshi has more than one way to approach as a main, and therefore can be unpredictable, and adapt much more mid-match than set-playstyle characters.
Yoshi can evolve from a defensive wall to an offensive tank seamlessly, with amazing aerial mobility and a relatively strong and annoying to stop projectile. His hitboxes are almost always a threat, and a few attributes such as double jump super armor, the ability to turn mid-air with B pivoting, and acceptable landing lag means he can be a tough character to break.
His offensive can be anywhere from rush-down to guerilla warfare hit-and-run mindgames, changing to what your opponent suffers with most.
However, I tend to disagree with NOT picking up a secondary. Besides learning more mechanics of the game you're actually playing, and having a back-up option if your opponent annihilates you, Yoshi has limits. Granted, these limits are not in the character, whose excellent ability to chameleon to playstyles works best with players who enjoy adaptation more than execution, but in the players limitations. See, it's simply human nature to fall back on what you're comfortable with. There will be a way you play Yoshi most often, and most comfortably, and eventually someone will beat you for specifically that.
No matter how hard we try, we can't overcome how we perform in split second decisions: not until you're a god of the game who is so well versed in every option that every option is the most comfortable. Until that point, pick up a character that plays differently: one that beats what you can't quite do with Yoshi, yet. It'll change your mindset a little, and break long formed habits with the character we love that will allow you to do more than you thought you could. Picking up another character is like bringing a little extra cash in your wallet, just in case you want an extra large popcorn at the movies, instead of a medium. It's just insurance.
Example: I have trouble beating rush-down characters with Yoshi. Fox, Sonic and Falcon are my worst nightmares. So I picked up Pikachu. I just couldn't see past the defensive control-related gamestyle I was most comfortable with. I picked a character that could stand up to rush-downs, so now not only am I more comfortable against that playstyle, but I rarely ever mis-input my egg toss for egg roll any longer.
Pick up Olimar, pick up Mario, pick up Pac-man or Ryu or Shulk or Roy. Pick a character that is for specific situations if you want, or pick up Sheik, pick up Rosalina or Zero Suit and become bilingual at the top of the tiers.
But pick up another character. Even if you never use them. Pick it up for your own sake.