I apologize in advance, but this is going to take a little bit of basic physics to understand the flaw.
“4) Rivers run down to sea-level finding the easiest course, North, South, East, West and all other intermediary directions over the Earth at the same time. If Earth were truly a spinning ball then many of these rivers would be impossibly flowing uphill, for example the Mississippi in its 3000 miles would have to ascend 11 miles before reaching the Gulf of Mexico.”
What the author is attempting to convey is that if the surface of the Earth is convex, then traveling around the surface would be akin to walking up hill.
Remember first that gravity is always pointing perpendicular to the surface of the Earth with roughly the same force (a little less when at higher elevation, a little more when at lower elevation, but within an imperceptible difference to people).
Forces are what cause water (or anything) to accelerate (or decelerate). Remember:
acceleration = Force ÷ mass
So the bigger a force is, the more acceleration it will provide an object, and the more massive an object is, the more force it will require to cause acceleration (inertia = an object at rest stays at rest or things don’t accelerate without force).
Since gravity is perpendicular to the surface of ball Earth, it will not make movement along the surface any easier or harder. Force or energy is required to bring something up to a higher level from the surface. This builds up something called Potential Energy. This potential energy lets things fall back to the ground from a height (or slide down a ramp or any other sloped incline like a river).
The same thing happens to cause rain where heat adds energy to water leading to evaporation and movement to a higher level of the atmosphere. Once the energy dissipates and the water cools to a liquid, the potential energy in the water is converted to kinetic energy and the water falls to ball Earth.
Even on a ball Earth, water still goes downhill.