I leave mine IN any quilt I'm not planning to hand quilt, here's why. When using this square-on-a-corner technique, it's easy to sew these squares across the diagonal, press to the corner, and then find you have a bit of overhang no matter how careful you were. When you cut off the background and the middle layer of the triangle, you remove the absolute square or rectangular cut of the base shape...and that makes it harder to line up the snowballed segment to the next block segment. Even if 'overhang' is there, you can match edges of the segments together well if you leave the background part of the 'triangle'.
Jan in VA