For example Rock Lee has a super speed when he takes his foot weights off. And without them he is very fast. And he doesn't even use chakra at that point. So that does happen in narutoverse.
Thats due to physical Nature thats what the weights were for, no charka involved and he has never been called the green flash or so fast that people thought he telported even with the gates ,Lee can't use ninjutsu to speed up so he has to use the gates which opens the limiters in the body so his charka can flow and makes him faster.Key word, limiters which do not allow the charka to speed you up unless you use a ninjutsu like body flicker.large amounts of charka does not mean you automatically are faster you have to know how to use it.
So no that does not happen in narutoverse.
When Naruto was able to take the foxes charka his physical nature would change too.So he was faster as his physcal nature changed,and never that fast.Now all he has got is the foxes charka, and if you don't know how to use it,it don't mean nothing.Bee was surprised, so that means he don't get that kinda speed from his charka and none of the other Jinchūriki either.Plus how can Naruto get that kinda speed from the foxes charka and the fox can't do it with is own charka???
So it don't exist at least FTG and body flicker are real in the narutoverse
It would be cool if Naruto could open the gates and the life energy from the foxes charka could heal him..