I'm usually careful when I chose my wording and try not to mix terms like "contactor" and "overload controller", as these have different meanings in my experience. I used ABB 3-pole and 4-pole contactors, which are not motor starters and do not have overload protection, to control single-phase equipment because I had them in my spare parts bin.
In normal operation, the single-phase equipment connects directly to the wall-mounted outlet and is switched on at the device (we don't have switched outlets). The contactor I used is essentially a 3 Form A or 4 Form A relay that is available in an easy to use and configure package for incorporation into my wiring scheme.
If I was going to configure a single phase motor control circuit from scratch, I would buy the correct components and connect them accordingly. While it's certainly possible for such an item to exist, I have never seen a thermal overload protection device fail to activate when any of the legs overload. I have more experience with Square D, Allen-Bradley, and Cutler-Hammer controllers, but in each case any leg that activates trips the controller.
When certifying equipment during installation, each leg must be tested individually to ensure a single phase overloading will deactivate the contactor. When used in single- or double-pole applications, the unused legs of the thermal overload device are along for the ride and do not adversely affect the operation of the controller. It might be different in the 450+ page regulation covering UK installations; however, my shop wiring was inspected and passed with no issues, including my single-phase 3-pole contactors.