I don't have evidence, but if you consider the flavor of the various Barbarian subclasses you can see that they seem to have a LOT more in common with the 4e Primal Barbarian than the 3e "Dumbass who gets angry" class. The 5e Barbarian owes a lot of its subclasses to the 4e Barbarian and Warden, not necessarily in terms of mechanic, but certainly in its lore. Even a bit of Shaman thrown in, too.
I don't think they would have been able to justify an entire class based solely on the 3.X version of the Barbarian. They probably would have had to plumb the depth of prestige classes and try to coble something together. I'm not versed in all the 3.X prestige class so I guess I can't really speak to that, but I do know the Primal Power Source, and the 5e Barbarian is Primal AF.
Obviously there is no way to know 100%, since in our reality 4e exist and a '5e' without 4e would have been WILDLY different without the lessons thought by 4e. For one thing, I think the way 4e basically burned brightly before crashing is responsible for the simpler design of 5e and its more sparse release schedule.
Personally, I believe that if the 4e Warlord was apprently did not have a 'strong enough identity' to exist as a class and was, supposedly, made into the Battlemaster Fighter, then there is no way in heck you can justify a 3.e Barbarian as a 5e class, aside from blind adherance to tradition.