After a total of about 3 days of practice with her harness, after which period she was performing nicely, I introduced the dogcart. I had fashioned a homemade lightweight rig from an old bike trailer. I took the bike trailer apart until all I had left were the wheels and the square, lightweight steel frame. I cut a piece of stout wire fencing/paneling and fitted it to the frame. Then I repurposed a pair of aluminum crutches for the shafts (notice that the emphasis is on lightweight).
The presence of springs on a horse-drawn carriage seems to be of little importance when judged by purely essential criteria. They are not needed to start or stop the vehicle. The horse provides motive power and brakes will stop the carriage when necessary. A carriage without springs could still function, as springs do not serve any absolutely necessary task. They do not light the way at night like lanterns do, or keep occupants dry in a rainstorm like a top. But riding in a carriage minus springs would be most uncomfortable, especially over cobblestone streets or on a country lane, or on a long journey.