As you've surmised, it is only for erasing pixels that you have painted on with the paint/draw tool in SC -- it won't erase original contents of the screenshot.
SC is not mean to be a full graphics editor, so more more complicated editing, you can use a real graphics editor like Gimp -- which you can set as a tool in SC to easily launch it, and SC will automatically pick up any changes and keep your annotations when updating.
With that said, SC does have a few specialty features for "removing" stuff -- under the SpecialFX1 menu you will find effects to blur and hide information in the selection region.
And under the SpecialFX2 menu you will find options to splice out areas of your image, OR "Patch" areas to make them equal to background -- it's this last option that might be most useful to you.
