It seems is no longer possible (it was in ArcMap) to Add field to Raster Attribute tables, except through the Add Field geoprocessing tool.
The Add field button in the table window is inactive and it shows the message "This command is currently unavailable. The project has unsaved edits or the table or its schema is read only".

My opinion is that in ArcGIS Pro ArcGIS raster table schema is inactive by default.
Can anyone confirm (or deny) this?
I'm using ArcGIS Pro 2.8 and it's the same in 2.9