No, such a tool generally does not exist in simple vector graphics tools
(such as Publisher's drawing tools).

Paint is a raster graphics tool (i.e. it deals in pixels, hence the
"distortion" you mention) and doesn't support transparency, which makes
an eraser trivial to implement. To make a working eraser for a vector
tool is much harder; Publisher's shapes are vectors (which is why they
are not pixellated when you print them), but Publisher's facilities are
not sophisticated enough to implement an eraser for them.

You need to rethink your strategy; what Mary offered you was an
alternative strategy to using an eraser. Instead of creating lots of
circles and then deleting bits, create lots of arcs - this is the same
as a circle, except that it is broken. Then you can adjust the handles
to show the bits of the circles you want.

There is no eraser in any Office drawing tools. Maybe the new PowerPoint??? Try
www.getpaint.net, it is free. It is not a vector application.

Download the free DrawPlus from Serif.
http://www.freeserifsoftware.com/software/DrawPlus/default.asp