Ok, I've finally come to understand that my repeated assertion that the falling observer in Schwarzschild and Lemaître coordinates needs to resynchronize his clocks was incorrect.
Let's approach the problem from the other side.
So, the falling observer does not need to resynchronize his clocks for the speed of light to remain isotropic. This means that the light cone actually tilts as one moves through a gravitational field. Only the coordinate system that accounts for this tilting in the necessary proportions is a physical coordinate system.
The key point to understand is that if we accelerate but do not need to resynchronize our clocks, it means that the light cone tilts. In special relativity, it's the opposite: when we accelerate, the light cone does not tilt, and it is necessary for the moving observer to resynchronize their clocks to maintain the isotropy of the speed of light.