"But everything falls apart once you try applying either theory to predict how three objects will move in the future based on their velocities and positions." -- That is simply not true. The three-body problem is simply hard to solve analytically (solutions exist only for some special cases) and is usually computed numerically.

Just to put this person's publications into context: Physics Essays is a journal with an impact factor significantly below 1, meaning that basically nobody cites its articles. And it is known to publish articles which would never make it through peer-review in more established journals. For Reports in Advances of Physical Sciences I couldn't event find an impact factor, which is not very reassuring.