I am not sure if you have come to your solution yet.
Regardless, I was able to export a feature layer as csv and retain expression values. This article gave me some guidance on the topic. https://doc.arcgis.com/en/arcgis-online/manage-data/calculate-fields.htm
In summary, for your attributes (intersect_1 | intersect_2 | intersect_3), you can apply an expression to the each column via SQL or Arcade Expression. This can be found in the data tab of your feature. Simply left click the column header and select "Calculate."
Using an Arcade expression will require that the feature is Sync disabled and update tracking disabled.
Using SQL will allow you to set an expression to all of the values of the column and continually update values as you continue to add data to your attributes. Sync enabled and data tracking.
Hope this is close to what you were looking for!