Further observations on the 'problem'. It seems that it is not necessary for that setting to remain at 'Sonic sounds for headphones'. It can subsequently be set back to 'Off', and the headphones will keep working. There is one slightly odd quirk that remains: If you have the headphones working with the setting 'Ssfhp' (saves typing), and you then set it back to 'Off', and then click the 'Test' button, you will hear a short sharp sound in the left earpiece for that very first click of the button. Subsequent clicks of 'Test' will work as expected. Setting back to 'Ssfhp' naturally continues to work, but then setting it back to 'Off' once again causes the first click of the Test button to fail. All of which is of little import. The real item of importance is that it has to be set away from the 'Off' condition just once. Up until today I would have clicked on 'Test' a hundred times or more while I tried all sorts of other things, all to no avail. Whether or not it is possible at the very start (new system) to simply change the setting to 'Ssfhp' then back to 'Off', or whether it is necessary to exit the 'Settings' app between changes is something I don't know. Whether reinstalling the system will revert it to the 'not working' state, I don't know. I suppose it is possible there is some sort of non-volatile-memory associated with the audio hardware that wouldn't be affected by a reinstall. Lots of possibilities, but it's still curious that there is the one-time failure of the 'Test' button after switching 'Spatial sound' back to 'Off'. I'm just happy that it's working now!  
If your headphones won't work on your new Inspiron 5320, don't call Dell, and don't call Microsoft. They both will say 'update the driver', and when that doesn't work then "Ok, we'll reinstall the system" (MS). Which, of course, also did nothing for the problem; but they were long gone by then. To cut a long story short, if you go to Settings/System/Sound and scroll down, there is a box labeled 'Spatial sound', with a default value of 'Off'. I had previously noticed it, but figured it was for people with external speakers etc etc. Wrong! Maybe that's what it is/was intended for, but it appears that a by-product of having it set to 'Off' is that your headphones won't work. The system (MaxxAudio) detects that you have plugged in headphones, and announces it in a little pop-up notification, and all the Diagnostics/Troubleshooters/Test-buttons will say that there is no-fault-found. It's just that your headphones won't work. Click on the drop-down selection next to the 'Off' state and select the (only) option 'Windows Sonic for Headphones'. Voila, all your problems are solved! Having got all of that out of my system, I still only feel a little better. I don't mind if Support can't sort out a significant problem (it can be hard enough on-site, let alone at a distance), but to be unaware of such a simple bug.... I would guess that it wasn't designed to work the way that it does, but until something is changed (probably the driver), beware.  
Fix Headphones-Not-Working Inspiron 5320