Do those USB ports work for anything else ?

Like a USB mouse/ thumb drive ?

USB ports work just fine (SSD). As mentioned, USB port recognizes/powers the webcam if another laptop is connected to the monitor via USB-C, but not when Dell XPS 15 9560 is connected.

Lets assume you are plugging the Cam to Monitor 

Any notification (sound) when webcam is unplugged and re-plugged ?

Any unknown (other) devices in Device manager, or Yellow exclamation marks ?

Any changes in Device and Printer (control panel) ?

but if I plug the cam directly to XPS15 9560, works well. Only not if it connects via Dell S2718D monitor.

Strange !!!!  

So monitor and cable are fine as working with another computer

XPS 9560 is fine as same USB ports working fine for other USB devices. 

Only thing I can conclude it to XPS 9560 not recognizing the drivers for that webcam. >> 

Any notification (sound) when webcam is unplugged and re-plugged 

Any unknown (other) devices in Device manager, or Yellow exclamation marks 

Any changes in Device and Printer (control panel) 

No sound.

No unknown devices in the Device manager - cam microphone gets recognized, but not the cam.

Cam does show up in "Devices and Printers"

cam microphone gets recognized : Well that makes things more interesting 

Cam does show up in "Devices and Printers":  Well than where is the problem ?

In which apps are you testing the camera ?? 

Zoom may be ?? 

Windows doesn't have an option of Default Camera, 

So are you manual changing the camera 

Camera is not recognized by Zoom, MS Teams, or the Windows Camera App. If I connect it to the other laptop as described, I am given an option to select it manually and can change between the Logitech and integrated cam.

I am about to give up 
can you snapshot the Device manager with Logitech cam showing in it ? some thing is missing. 

after many plug ins and outs, camera is now recognized. mystery remains.

I did not update anything, nor the cam has any drivers to begin with. perplexing.