I wonder why JOSM does not recognise .kml files. Could it? If so, might it be in prospect?
Apparently at least some .kml files can be opened in JOSM if you install the Open Data plug-in: https://wiki.openstreetmap.org/wiki/JOSM/Plugins/OpenData#KML.2FKMZ_files
I thank you for your immediate and helpful reply; the opendata plug-in does allow me to open my .kml files. I have discovered recently how much better they are than .gpx because they retain their attributes in my Locus Map app and now JOSM, which is so useful.
I was always able to open *.kml with this plugin. Seems your files are special. Can you post a link to one?
The way in which I wrote my reply confuses, clearly: I shall state in another way, I am able to open my .kml files with the opendata plug-in; I am satisfied, totally.