When I try to add a Layer file (.lyr) I get a red exclamation point asking me to click to "repair layer's data source". When i bring this same layer file in the ArcMap 10.3, it displays the data/database that the layer file is referencing just fine. Does ArcMap Pro not support .lyr files? I would find this strange as .lyr files appear as a supported file type when browsing for data to add to a project.
Did you see the screenshot before you posted it? I see your Inbox and other stuff. Trimming the screenshot to relevant parts makes it easier to work with.
Anyhow, yes, layer files are supported, but I think you are looking at this from the wrong perspective. Layer files reference data sources, so are you sure the actual data source is supported? Are you working with a personal geodatabases possibly?