Good thing I'm not trying to sell anything! Analogously, if people don't want to escape the wilderness because I only showed them the path to the exit, rather than hold their hand all the way out, I won't try to change their minds.

This means that your velocity formula, while it may be correct in the new metric, should also be defined for the new range of r [; -r_s < r < \infty ;]

I don't understand why you think further definition is needed. I re-read your post but still don't understand "all you had done was to change the range of the variable r". Can you elaborate? What does a negative r represent? How can a body have a negative radius?

Intuitively speaking, as long as the curvature factors (the differences between the metrics for flat spacetime and Schwarzschild geometry) are smooth across the range of r, the metric should approximate SR locally. This is like saying that any sufficiently local patch of ground on Earth should be approximately flat as long as the surface is smooth (no cliffs or curbs allowed). I can't prove that, but I'm confident of it.

On how the Schwarzschild metric is inconsistent with GR as a whole, it's a subtle thought experiment. I'd walk you through the prerequisites and steps so I can rebut as objections arise. But first let's resolve the issue above, please.