Why does the screen remain blank after booting my Pi while its monitor is turned off/unplugged?

Because, unlike x86-based PCs, the Raspberry Pi does not support monitor hotplug (i.e. changes to the video configuration after boot-up).  This is not a failing of TLXOS, it's limitation of the Raspberry Pi firmware.

By default, if the Pi doesn't detect an HDMI signal on a port when it is powered on, then it marks that output as absent, and the O/S will be unable to activate it after boot-up.  If no HDMI ports are active, then the O/S will not be able to start an X (graphics) server at all, and will remain in TTY-only mode.

If your Pis are deployed in places or situations where the monitor might not be powered on before the Pi is, or does not exist at all ("headless", e.g. remote VNC access only), then you will have to force one or more HDMI outputs on and force usage of a specific resolution.  That is, the Pi must be instructed to ignore EDID information provided by the monitor regarding what resolutions it supports, and attempt to use a specific resolution anyway.

To do this, you must first use TMS or Tlxconfig to select a resolution other than "Auto" on the relevant video output, and then use "TMS' Device->Local Configuration->Edit Raspberry Pi Configuration" option to add the following lines to /boot/config.txt:

hdmi_force_hotplug=1
hdmi_force_mode=1

If your monitor is plugged into the HDMI1 jack rather than the HDMI0 jack (HDMI0 is the one closest to the USB-C jack on the Pi 4, and the one furthest from it on the Pi 400), add these lines instead:

hdmi_force_hotplug:1=1
hdmi_force_mode:1=1

These variables are somewhat misleadingly named, being engineering terms from the HDMI protocol perspective.  From the Raspberry Pi perspective, they actually relate to coldplug (not hotplug) events.