ArcGIS Desktop since version 10.0 was designed to create metadata in the ArcGIS metadata format. If you created ArcGIS metadata in any 10.x version of ArcGIS Desktop, that metadata is automatically available and fully functional in ArcGIS Pro and no upgrade is needed.
In the original question, the reason why the full metadata content could not be seen or exported in ArcGIS Pro is because the full metadata content is stored in the FGDC CSDGM XML format. ArcGIS Pro only displays, edits, manages, and exports metadata stored in the ArcGIS metadata format. This behavior is by design. See the View and edit metadata help topic, specifically the section "View standard-format metadata content".
In ArcGIS Pro, if only FGDC CSDGM-format metadata content is present as the item's metadata, no metadata content will be visible and the Edit and Export buttons are not available at all. However, ArcGIS Desktop allowed FGDC CSDGM-format metadata content to coexist with ArcGIS metadata. It is likely a small amount of ArcGIS-format metadata content is present in the item's metadata, such as a geoprocessing history entry that describes how the item was created. The presence of this content can tell ArcGIS Pro that ArcGIS metadata is available. Only the ArcGIS metadata format content is visible in Pro, while the more complete metadata content stored in the FGDC CSDGM XML format is invisible and unavailable. In this way, various metadata operations might be available to you in ArcGIS Pro, but those operations will only use information present in the ArcGIS metadata format. Since the full metadata content that you wanted to use is stored in the FGDC CSDGM XML format, that content is not be available to any ArcGIS Pro metadata operations.
The recommended solution is to do a full upgrade of the item's FGDC CSDGM-format metadata content to the ArcGIS metadata format. This transfers the full metadata content to the ArcGIS metadata format and makes all that content available in ArcGIS Pro. This can be accomplished in ArcGIS Desktop using the Upgrade button in the metadata display or using the Upgrade Metadata geoprocessing tool. It can also be accomplished in ArcGIS Pro from the Catalog view using the Upgrade > FGDC CSDGM Content command on the Catalog tab on the ribbon, in the Metadata group. Alternatively you can use a Python script to upgrade metadata for one or more items using the upgrade method in the ArcPy Metadata module.
import arcpy
from arcpy import metadata as md
archive_item_path = r'Data_with_FgdcCsdgm_notUpgraded.shp'
archive_item_md = md.Metadata(archive_item_path)
archive_item_md.upgrade('FGDC_CSDGM')
archive_item_md.save()
Even though ArcGIS Desktop 10.0 and newer were designed to create metadata in the ArcGIS format, accommodations were made to allow some FGDC CSDGM-format content to be displayed, if it was present. This facilitated migration from ArcGIS Desktop 8.x and 9.x that natively managed some metadata content in the FGDC CSDGM XML format. This includes the ability to see FGDC CSDGM-format title, theme keywords, purpose, description, credits, and use limitations directly in the metadata display. The workflow suggested in the Reply only does a partial upgrade to the ArcGIS metadata format--only the FGDC CSDGM-format metadata content that appears automatically at the top of the metadata display is converted.
Once any type of upgrade has occurred or any ArcGIS metadata format content is added, the Upgrade options are no longer be available directly in the ArcGIS Desktop and ArcGIS Pro applications. It is possible to do a second upgrade that transfers all FGDC CSDGM-format metadata content to the ArcGIS metadata format, but this process will overwrite any information that was typed using the ArcGIS metadata editor. Therefore, upgrading a second time can only be accomplished using the ArcGIS Desktop Upgrade Metadata geoprocessing tool, or by running a script in ArcGIS Pro that uses the ArcPy Metadata upgrade method.
This means if you followed the method suggested in the reply, it is then more difficult to do a full upgrade and have all of the original FGDC CSDGM-formatted metadata content available to you in the ArcGIS metadata format. The best practice is to convert all FGDC CSDGM-formatted metadata content to the ArcGIS metadata format the first time using the commands available in the applications.
A similar Esri Community post addressed the same general issue via a different question. My colleague Jill provided a link in her response to a video of her UC tech workshop, ArcGIS: A Practical Approach to Metadata, Catalog, and Search. This resource may help people who used metadata workflows in ArcGIS Desktop that stored metadata in the FGDC CSDGM-format to make the transition to ArcGIS Pro, where storing metadata in the ArcGIS metadata format is a requirement.