"But on closer inspection, his new shape could morph into a sphere (as Poincaré insists it must be able to do), but - remarkably - it could not do so smoothly. So, although it was topologically a sphere, in differential terms it was not."
Wait. I am certain that Poincaré himself never said a word about what is now called the "generalized Poincaré conjecture". His conjecture asked only whether a simply connected, compact 3-manifold is necessarily homeomorphic to the 3-sphere. (And in fact he expressed this solely as a question, without asserting that he believed it to be true.)
So, Poincaré never insisted anything at all about what a 4-connected 7-manifold might be able to do.
In fact, a little patient googling shows that the first statement of the generalized Poincaré conjecture (namely, that any compact n-manifold that is homotopy equivalent to an n-sphere is in fact homeomorphic to it) was given by the great topologist Witold Hurewicz in the mid-1930s.