Quick question - so it is just the Layer files (.lyr) that are the "incorrect" layer names as they have the path included?
Something to check - any chance the layer names in the original mxd were renamed in the Table of Contents, i.e. what is viewed there does not represent what the layer name is on disk? It is pretty common for folks to rename the layer name in the Table of Contents to provide a clearer explanation of what a layer is. For example, naip6_02_1 may not mean much to most people, but if rename it to 2016 Placer County Aerial a viewer will know what it is.
Check the Properties (Right-click on layer name in Table of Contents, Properties, Source tab) to confirm that the feature class name in the source is the same as what is in the Table of Contents.
Chris Donohue, GISP