You have all of the right words, just in the wrong places
Ok, I am going to do my best without getting too wordy. (using the above pic as your refernce)
The collar has rings on the hames that are hard connected to the trances (they are sewn on, no buckle). Which is connected to the singletree to the people mover and is adjustable via the chain links. Horses PUSH into the collar which engages the traces, which then pulls the people mover or carriage, etc.
Continuing with the collar, there is no false martingale (a strap of leather that runs between the front legs to the collar, because it is not needed, the horse is not pushing the people mover backwards, the false martingale stops the collar from going over the horses head). This leads us to no breeching.
Think of breeching like brakes. The horse/pony wears breeching to control the speed down hill and it holdsback the carriage/cart/people mover from running up the horse (see 2Dogs remark about the holdback strap).
Also absent is the saddle or backpad, back strap, and crupper and sometimes trace carriers.
These are not needed in this set up. Shafts are not needed either. (shafts are connected to the axel of the carriage)
The shafts turn the carriage. Notice that this people mover is on tracks so no need to turn or steer the people mover, it does it on the rail. All the horse has to do is walk forward and be directed, the mover follows. (and given its location, its most likely straight with turns at the end to go the other direction)
There is no breeching because the people mover has a break and a parking break on board which is controlled by the driver. Since there is no saddle or backpad, (which in traditional single driving would have the tugs which carries the shafts; with a pair or fours, the traces are connected to the collar and buckled to the saddle/backpad to keep things organized and not tangled up) along with a false girth.
The breeching is breaks. Yes they attached to the shafts so it holds the carriage back and not runs over the horse. Breeching like 2Dogs stated is connected to the shafts via the hold back strap, (looped loosely) around a shaft or sometimes there is a quick release snap.
There are no trace carriers because there is no back strap. The trace carrier is a strap of leather that runs from the trace over the hip and to the other side, and it carries the trace. In this situation its not needed. Think about the terrian, the gait the horse uses, and the speed, and that there is a break on the people mover.
Horses used in this type of work know their job. They know standing still means rest, and to get out of the collar, (which deactivates the traces) and not work. When they walk off the traces are engaged and are out of the way of getting stepped on.
In some harnesses there is breeching that connects up to the collar via the saddle/backstrap.
Also something to consider, they are using horses with good level brains. They know their job. Walk and then stop.