What is the best way to smooth a path?

I wish Adobe Illustrator would give users the option to delete extraneous anchor points on paths (such as those that occur in auto-tracing) without deleting parts of the path itself. The "simplify" option is an arguably crude tool that affects the path on a global basis. Sometimes only a small portion of a path needs clean-up work. I can't delete errant points on a path with the Direct Selection tool without deleting segments of the path.

One of Illustrator's rivals, CorelDRAW, has a similar "simplify" command to clean up junk points on a path. However, one can use the "shape" tool to individually select anchor points and delete them. When the points are deleted the overall path still remains. CorelDRAW does its best to maintain the original shape of the path, but with junk points deleted the resulting path will be more smooth. If parts of the path itself need to be deleted the shape tool can be used to quickly break apart the path via selected anchor points.