Paste Special does not require selecting the layer in the Contents pane. Selecting a layer from the Contents pane takes focus away from the map and gives it to the Contents pane. The Contents pane allows the copying and pasting of layers, not features. To copy and paste features, the focus must be on the map. Refer to the following documentation for additional information: https://pro.arcgis.com/en/pro-app/latest/help/editing/paste-special.htm.
The issue of the 'Enter" key not working within the Paste Special dialog is fixed in ArcGIS Pro 3.2.