Can't even begin to agree with that assertion. 5e uses full-on straightjacket classes, just like 1e, really. It curbs that slightly with 2e-Kit-like backgrounds, and optional 3e-lite feats & MCing. Sub-classes (archetypes, BTW, is the label for Fighter/Rouge sub-classes, every other class has a different term for it, just say'n) represent a level of flexibility somewhere between old-school hardwired sub-classes that were just alternate classes and 2e specialty priests/specialist wizards, OT1H, and 3e/4e 'builds' OTO - as befits a compromise edition, I suppose.
But, it remains very much - as it's meant to - traditional D&D, which means it's just a long lists of long lists (though less long than any ed since 1e), with playing the character you want ultimately dependent on there being something 'close enough' on the list.
3e & 4e were both had much more flexible rulesets than 5e. Much more. Not even a fair comparison, really.
What flexibility 5e is an aftermarket accessory added by any DM willing to wield a machete. It's like 0e/1e, that way, really, which makes it a fine tribute to the early game, a perfect vehicle for the current come-back, and a laudable marketing/commercial success.
Sub-classes are more like little mini-classes unto themselves than builds in 3e & 4e or Kits in 2e, and less like that then sub-classes in 1e (which were, totally, full classes onto themselves, just inheriting a few things from the nominal over-class, like what magic items they could use, for instance).
So, really, 5e 'needs fewer classes' in the sense that it can proliferate sub-classes to the same basic effect as proliferating classes (especially PrCs).
It would not have been that hard to reduce the number of classes to 4, rather like the class groups in 2e, and have Paladins, Rangers, & Barbarians (& Warlord) sub-classes of Fighter, Sorcerer & Warlock (& Artificer) sub-classes of wizard, Bard & Monk sub-classes of Rogue, and the Druid (& Avenger, & Archivist) as sub-classes of Cleric.
We'd still need a freak'n Psionicist, though.