You can use the lasso for removing areas, which is a great and super useful alternative to the erasers (as you are saying).
But in that case we call it the Cutter and not the Lasso. Just hit X instead of L – and you will be able to draw the dotted line and pick up the area as you are used to with the Lasso. The only difference from the Lasso is that the area is removed from the drawing. Now you can stamp it down elsewhere, but if you don’t need it, just hit Esc to loose the cutout.
So to remove an area from a drawing: Hit X, mark the area, hit Esc.