I wonder if you have to use the direct selection tool ( white arrow ). I'd just delete the extra anchor points manually, you'll have more control that way.
Whether anything happens depends on the settings you apply when you Object>Path>Simplify; there is also the Smooth Tool.
But the best results will be obtained by creating a smooth path from the start.
You may redraw with the Pen Tool, following the angular path.
Where is the smooth tool.
I tried the direct select tool (white arrow).
When I clicked on a single point to delete
to smooth things out the section between
the adjoining points disappeared after I selected
the dot to delete and hit delete.
The Smooth Tool is found as a flyout of the Pencil Tool in the Toolbox.
You remove Anchor Points with the Delete Anchor Point Tool (flyout from the Pen Tool), but with Corner Points on either side you just get a straigh Segment.
The idea of the simplify command is not to alter what you have but to make it less complex.
But there are options and settings if you take a closert look.
The smooth tool might be more instintive it is under either the pencil tool or some other tool but it is under one of the other drawing or transformational tool.
You might fin the reshape tool helpful as well under the eraser tool, but the reshape tool is not a smoothing tool.
I thnk the smoothing tool is something you might find more handy.