Your points reverse direction and perhaps are out of order:
ParametricPlot[{line[[1]][tt], line[[2]][tt]}, {tt, 0.2, 0.3},
Epilog -> {Point[points[[All, 1 ;; 2]]]}]
Your points reverse direction and perhaps are out of order:
ParametricPlot[{line[[1]][tt], line[[2]][tt]}, {tt, 0.2, 0.3},
Epilog -> {Point[points[[All, 1 ;; 2]]]}]