Clearly in SC coordinates r is an affine parameter, not t. I never made a restriction to specific coordinates. Obviously a change of coordinates must be introduced for the spatial part of the metric that puts it in conformally flat form(spatial part that in the case of the statiic Schwarzschild metric is made equal to the time part in the case of null geodesics), wich is perfectly valid, now you are claiming that particular coordinates are fundamental?Actually, that shows that coordinate time is NOT the affine parameter. Changing from the affine parameter to coordinate time as an independent (not affine) parameter allows the treatment of static gravity as an optical medium. It doesn't change that the affine parameter (for all types of geodesics) is unique up to linear function (affine parameter means the simple - parallel transport - form of geodesic equation must be satisfied using the parameter). Thus, in SC coordinates r is an affine parameter and t is not an affine parameter.
In general such conformal tranformations change the form of the geodesic equation and one doesn't obtain a geodesic, but the exception are null geodesics, but then the parameter is no longer affine, again static spacetimes are an exception to that and one can recover the affine parameter, in this case the coordinate t, after the transformation to a conformally flat spatial part.
Quoting once more from Padmanabhan page 160:
"....This is the same as a geodesic equation with an afﬁne parameter t in a three-dimensional space with metric Hαβ. It follows that the null geodesics in a static spacetime can be obtained from the extremum principle for coordinate time δ ∫dt =0 , (4.79) "