However, I remembered that the one I removed had most of the tubing coiled up inside the reefer. Is that they way they're supposed to be installed?
To give you a definitive answer you really do need to provided the make and model of the thermostat. Some have a fixed "span" of a few degrees and others are adjustable over a huge range either side of the set point. Changing the setpoint alone may just move the upper or lower span limits out of range of the* machine.
In the meantime ...
It really doesn't matter where the excess capillary is located as long as it is not kinked. If it was all inside the box then replace it the same way.
Make sure the capillary or sensor bulb is not touching the cold plate or the sides of the box. It needs to detect the air temperature, not the box or its contents. Make sure air flow around the bulb or tube is not blocked.