No, they can't. One could very well come up with some other coordinate system whose time & space coordinates are (non-linear) combinations of the usual Schwarzs

No, they can't. One could very well come up with some other coordinate system whose time & space coordinates are (non-linear) combinations of the usual Schwarzschild time & space coordinates. In that coordinate system, spatial slices of equal time would be different, so spatial distances would be measured differently. TL;DR Just pretend the author wrote "[…] equally spaced with respect to some observer's frame of reference" (e.g. a stationary observer at infinity that uses the usual Schwarzschild coordinates).