I think the idea of the white man giving the tipi to the plains NDNs is a stretch, but tipis (lavvu) were used by nomadic Europeans (Samis, or Laplanders) before America was discovered, so is could be possible, I suppose.
Before they had horses, the tipi would not have been a very suitable nomadic dwelling. It would have taken a very large pack of very large dogs to move one, think over a dozen full thickness large buffalo hides and 17-21 poles over 20' long. The poles would be possible, to some extent, but the cover would really be impossible to move by dog, I would think. It is obvious the lifestyle of the nomadic tribes evolved quickly after the introduction of the horse, which started in the 16th century, and was pretty widespread by the late 17th century. That means there was 200 years, or 7-10 generations for them to develop into the plains NDNs we think of.
BTW, I have an 18' tipi, and it is really a semi-permanent dwelling, when you think about it. A wedge, or just about anything you can conceive out of a rectangular piece of canvas is far more practical to move from place to place.