That seems like a reach when he added 3 more via DLC and even more in Ultimate through DLC while keeping everyone. Sakurai has gone on record stating multiple times that Nintendo makes the decision for who gets added not him, so him limiting the 3rd Parties to those 3 doesn’t make a whole lot of sense. The decision almost certainly had to come from either an issue with rights, time constraints, or from another source at Nintendo whoever that is.
The ground had to be tested somewhere though. Sonic wasn't a too risky third party character to add, Snake kinda was, and Sakurai didn't exactly wanna go overboard apparently. The next up third party characters in Smash 4 where also quite "safe" additions. Mega Man and Pacman, with Ryu as DLC and later on Cloud and I guess Bayonetta. I also don't think Ryu is extremely risky, and Bayonetta was already becoming Nintendo exclusive around that point.
Cloud also was just about the most requested character whom many perceived as "never gonna happen", so that was a huge factor. I remember support around the time of Brawl for Cloud being enormous, but always shrugged off as unlikely.
Also Brawl was delayed to push Sonic in, he wasn't a high priority addition probably therefore. It might as well just have been Snake.
Come to think of it. What were the characters confirmed to do well on the ballot?
If I remember correctly...
Sora
Banjo and Kazooie
Snake
Veterans in General (though Snake was specifically named)
King K. Rool
Ridley
Chrom (in the East)
Dark Samus (in the West)
Bayonetta (apparently she did really well in Europe)
Simon (Castlevania as a whole seemed to do well)
Cloud (most popular Final Fantasy choice)
Dante, Lloyd, Shantae and Dragonborn (confirmed to be popular choices / not sure if this counts)
Did I miss any?
Inklings and Isabelle? I know many thought of them as inevitable, and I really think they where. I predicted them both too. But Splatoon was huge during the ballot, and Villager's inclusion naturally must've given Isabelle tons of attention. Both also got DLC Mii costumes in Smash 4 I think? Seeing that K.Rool got the same treatment, that's probably a hint. I think
Daisy is another one that probably did very well.
Also from the missing veterans, many must've gone to Wolf, Ice Climbers and Pokemon Trainer. I remember those 3 being missed by many too. And I remember many people claimed they voted for then.
Another I wouldn't be surprised about at all is Sephiroth. Cloud got in before the ballot, I imagine most of the fans of Cloud pushed for Sephiroth afterwards, much like how the K.Rool fanbase was the same crowd that rallied behind Diddy before.