It is great to hear you are documenting at the field level. I do this regularly. Not enough people are documenting the kind of information one should be adding at the field level.
So initially I have to admit I thought you must be doing something wrong). Turns out you are not but tracking down the issue took me some time.
So first of all ArcGIS Pro's ability to handle metadata is an odd one, I think the interface for documenting metadata is much improved but the only tools to manipulate metadata are on the tab you having been using in the GUI ribbon. There is a complete lack of metadata tools for the geoprocessing toolbox, not sure why but hopefully something ESRI will resolve. So far the fudge that ESRI have implemented is to expose a metadata module in the python environment, would be nice if these were actually polished tools in toolbox so they could be used in ModelBuilder.... One can only but hope.
Assuming ArcGIS Pro was just being flaky I went back to Good old never lets me down ArcCatalog and to my surprise was able to replicate the same issue when exporting the metadata! All that important information stored at the field level was being dropped.
So looking at the metadata before any export, any field information you have judiciously added sits in the
<eainfo> node of the XML hierarchy.
A quick search for
<eainfo> ended up at the FDGC web page. I then read the ArcGIS Pro help file on the export tool and that page even says:
Some ArcGIS metadata elements, such as the thumbnail, are not exported
because they don't have a corresponding metadata element.
So using ArcCatalog and exporting the metadata using the translator
ARCGIS2FGDC.xml I was able to preserve the field information.
It seems to me that only the FGDC standard actually honours the information stored at field level. Without looking I can only assume that the ISO 19139 standard either does not store that level of information or the translator does not map to the equivalent space in the ISO 19139 format so the information never gets exported?
So if you want to get your field information out it appears you need to be exporting to FDGC standard. May be you can get such information out into an ISO compliant metadata file but I assume you would have to create your own translator and that's outside my skill set to comment on.