Hi Fiona,
This is indeed something that has not yet been implemented in ArcGIS Pro. I know that there is an existing enhancement request that is currently being reviewed by our development team.
ENH-000126352: For ArcGIS Pro include a equivalent GUI interface to View cache completion status that is available from a service via a service connection.
If you have access to Technical Support, please log a case and we can get you attached to this enhancement so that you can monitor its implementation status.
Additionally, you can submit an Ideas post for this as well: ArcGIS Ideas
For now, the cache status can be viewed in the following ways:
-ArcGIS Server Manager
-ArcMap/ArcCatalog
-As the result of the Manage Map Server Cache Status tool Manage Map Server Cache Status (Server)—ArcGIS Pro | Documentation
Calvin