There's a related enhancement request from June 5, 2020 asking for this functionality that I think is very similar to what's being requested here. If an ArcGIS Pro project is already open, it adds the *.lyrx to an open Map View. Else if AGP is not open, it starts an AGP and adds the *.lyrx to a new Map View. Current AGP workflow is double-clicking a *.lyrx file in File Explorer, it starts a new AGP project regardless of an open AGP session or not.
ENH-000131830 : Requesting the ability to open the .lyrx file in the same session of ArcGIS Pro when clicking on the .lyrx hyperlink from the identify tool window.