XPS 13 (9370) 3 external display issue

Here is my guess as to what's happening.  The D6000's documentation indicates that when a DisplayLink driver isn't present, the HDMI port gets used in regular DisplayPort Alt Mode, i.e. the D6000 taps into the native GPU output wired to your system's USB-C port in order to drive the HDMI output.  The fact that your TV is connected via that HDMI output and lights up when you close the lid, while your other displays (which require the DisplayLink driver) are no longer even detected doesn't seem like a coincidence.  The question then is why the DisplayLink driver is shutting down or otherwise becoming unavailable when you close the lid, even as the GPU remains available to drive the HDMI display directly.  My only guess as to what may have changed here is a driver or even a firmware update having been pushed via Dell Update or even Windows Update.  I've read that some Dell systems are receiving BIOS updates through Windows Update these days, for example, and maybe the newer version of that driver/update introduced this behavior, intentionally or otherwise.  You might want to post on the DisplayLink forums to gain access to an audience more knowledgeable about this specific technology.  They might have a way to keep the driver running even with the lid closed, or might at least be able to confirm what's going on.