Is there any advantage to having equipment connected to a UPS device by simply running piggyback off the device?

Not too sure what you mean by 'running piggyback' but UPS devices are purely intended to go between the mains supply & your system in order to (1) provide protection from spikes, brownouts etc. (2) provide a temporary (usually only up to 10 minutes or so) power supply to allow you time to shut down if you get a blackout or (3) automatically set Windows to shutdown automatically if the UPS cuts in. UPS devices normally only power the system & the screen with a USB connection for the Windows control. Putting extra devices on just means that the UPS power time will be reduced. "I know that I'm mad - I've always been mad..."

The UPS was set up by an outside service and NO Auto-Shutdown was configured. The unit simply has several devices attached to it as a very temporary alternate power source to the wall outlet-very basic. Power config in Control Panel shows connectivity via Com ports 1-4 but no serial cables are in place so the Config box is unchecked.

Pt Cruiser: If there is no serial cable or usb cable connected from the computer to the UPS, the ups cannot auto shutdown the computer. When the battery runs down the computer will just abruptly shutdown, as if you pulled the plug, yourself, or if there is no automatic voltage regulation (AVR) it may go into a brownout mode first. Both are bad for your computer, can cause data corruption, and in the case of brownouts, may damage internal components.