Ya, I actually just got to that last part. Had a "that escalated quickly" moment.
One of the main problems I have are his projectile motion diagrams (throwing a ball straight up on a moving train or firing a gun straight upwards). He don't take Newtons first law into consideration, and assume that the force would be "used up" after the ball reached its highest position.
When drawing a force body diagram, you can treat horizontal and vertical motion separately, so while the upwards horizontal momentum would eventually be "drained away" and reversed by gravity. There are no forces acting in the vertical direction except for air resistance. Experimentally, you can walk forward and toss a ball straight into the air. It will continue moving forward until you catch it again (or drop it on the ground).
This principal, that projectiles (any particle moving with both vertical and horizontal vector components) follows a parabolic path (rather than something that looks more like a right triangle as illustrated on page 66 of "earth not a globe...") has been experimentally demonstrated ad nauseum.
If you were standing on a train and throwing a ball up in the air, it wouldn't surprise me if the higher air resistance (wind from traveling at 30 mph) would cause the kinds of results that Samuel Rowbotham got.
The same principle applies to the experiment involving firing a gun straight upwards. The only difference is that in RET, the atmosphere rotates at the same speed as the planet, so vertical air resistance would be negligible unless it was windy.