The Copy and Paste commands in ArcGIS Pro are contextual. That means that if you Copy features the map must be in focus (highlighted) when you Paste them. The issue is that the focus returns to the Tasks pane when you press Run or Next.
But...
There is an easy workaround: use the geoprocessing tool Append (Data Management) to append the selected features into the target feature class. You might want to add a verification action to make sure there is a valid selection (more than one feature but less than X) - this way you will not append all the features if the selection is empty.
