I would like to ask how rigorous is the statement that Schwarzschild metric has coordinate singularity at Schwarzschild radius. The argument is that singularity at Schwarzschild radius appears because of bad choice of coordinates and can be removed by different choice of coordinates. However removable singularity has closed neighborhood around it. But singularity surface at Schwarzschild radius is open toward infinite future and infinite past. Then if we consider gravitating body that has formed at finite past it would be open toward infinite future only. And considering that Schwarzschild metric is vacuum solution in such a case we have to glue it to some other metric that has non-zero stress-energy tensor. So this coordinate singularity has to first appear in some non-vacuum solution and only then it can be glued to Schwarzschild metric's illusory singularity. So in case of gravitating body with finite past the argument seems to depend on existence of such valid non-vacuum solution. Another argument is that coordinate independent quantities are finite at Schwarzschild radius so it should not be real. But that argument can be turned upside down and stated that coordinate independent quantities should not approach finite values as space-time is extended toward timelike infinity. And taken way the argument is less obvious.