While there doesn't appear to be a "Reset Shape" option that will quickly reset a shape's connector points, I've found that the Change Shape option actually accomplishes the same thing.
Simply select all shapes where you'd like to reset the Connector Points, go to Change Shape and just select the same shape. Shape layout and content will be preserved but the Connector Points will be reset to the default.
Obviously you'll need to do this for each type of shape separately but it seems to be the fastest way to reset Connector Points.
When it comes to removing all Connector Points, this actually is possible as well. The default Rectangle drawn from Tools has no Connector Points but of course doesn't show up in the Change Shape menu. However, you can drag any Shape you like from the main area into the Document Stencil in the Shapes panel on the left. If you then jump to "Shapes in this Document" in the Change Shape menu, you can use that base shape which will remove the Connector Points.
This also allows you to essentially apply a Connector Point template to shapes by dragging the finished shape to the Document Stencil. Do note that this is best done by modifying the Connector Points from the base stencils in the Shapes panel. The shapes drawn from Tools appear to have some non-standard behaviour when used in this way.
Note: the Change Shape feature was apparently introduced in Visio 2013.