We are also having this issue and I think I figured out a workaround, but there seems to be something inherently off with Pro's ability to work with metadata in any format besides the ArcGIS metadata style. It would be really great if Pro had more ability to work with other types of metadata styles as the standard my agency is supposed to be using is ISO 19115-3, but if you export metadata as any other option than the template and try to read it in Pro, it will give you an error message that because it is in one of these formats it can't be displayed or edited in ArcGIS Pro (example image uploaded). This is really confusing since there are multiple options in Pro for the metadata styles. We have discovered that metadata can be very tricky in Pro because there's options to only save it within the Map area of the .aprx, which will not be able to be viewed on the source data from another .aprx.
Here's the workaround that seems to be working for me to get metadata to display on .lyrx files, in Catalog View.
1) Save the original source layer metadata as a Metadata Template (this must save it as ArcGIS format, see image). Make sure this .xml saves to the same area as the layer file or move it there.
2) Using File Explorer, ename the .xml to the exact name as the .lyrx file including adding the.lyrx to the end of the file name
3) Refresh in Catalog View and then the metadata should display.
Importing the file in the Catalog View whether as an exported xml or the metadata template, which is presumably in the ArcGIS metadata format, does not seem to work for me consistently. This is fairly similar to the work around I do in ArcCatalog with .lyr files and is actually at least one less step. But at least in ArcMap, .lyr files will typically pull at least a title and item description into the metadata from the source layer and in Pro nothing is carried over.
