Publishing map from ArcGIS Pro to ArcGIS Server as Map Service?

You cannot publish a map service to ArcGIS Server from ArcGIS Pro. There are no GP tools for that available in Pro and no GUI options for that either. No arcpy functions that could do that either. As you already found out, your options are ArcGIS Online and Portal.

A quote from an Esri blog post in February 2015:

The first release of ArcGIS Pro does not allow you to publish geoprocessing services. In fact, you cannot publish any services to an ArcGIS server, as you can only make user connections to these servers. Publishing functionality will be added at a later release.

In spite of that, it might be possible to publish Geocode, Geoprocessing, and Image services. This is because there corresponding functions in arcpy available for creating drafts in Pro for that. Then one needs to use the Stage service GP tool and then Upload service definition (both of them available under Server Tools > Publishing). I haven't tested that yet; it might be that those tools are made available for the future releases, but it might just work as the Python code doesn't seem to differ from the standard ArcGIS Desktop.

I don't work with Portal, but when you share a hosted map layer in ArcGIS Pro to Portal, it seems as it should end up on the federated ArcGIS Server. So, it might be a way to go if you need to make your resource available to non-Portal users with ArcGIS Server interface.

The functionality to publish map services via Python using CreateSharingDraft has been implemented in version ArcGIS Pro 2.3:

Creates a MapServiceDraft from a Map in an ArcGIS Pro project.