Understanding of how the trigger connector works on a boom-arm vehicle barrier
When the remote is pressed the trg pin is then grounded.

This would be a standard implementation. A weak pull-up resistor pulls the input normally high. A ground connection will pull it down and overcome the pull-up.

From this manual, they say the boom barrier is &quot;Triggered through a normally open dry contact&quot;.

&quot;Dry&quot; here means that the contact is isolated from whatever is controlling the contact and no voltage appears on it from the driving circuitry. It is a standard interfacing technique to allow systems working on differing voltages to interact with each other without damage or complexity.
If you are adding another trigger then the simplest thing is to use a relay contact or a pushbutton to switch the controller.