Related: https://community.esri.com/t5/arcgis-pro-ideas/need-absolute-path-names-in-pro/idc-p/1569417
Pro 3.1
Give us the option to save with absolute paths when using "Sharing--> Save as Layer File"

This is available through the geoprocessing tool Save To Layer File (Data Management)—ArcGIS Pro | Documentation

But is unavailable through "Save As Layer File"

This discrepancy causes issues: LYRX files are meant to be easily moved around and re-organized, and being tied to relative paths means that they have to be re-created manually (Since we still can't edit the LYRX in Catalog). Additionally, most users do not use the geoprocessing tool to create LYRXs, meaning that the majority of our LYRX end up being created using relative paths.