I've worked with some GPS tracking hardware, and they're usually much bulkier and require much larger batteries than a 2050 cell. If I had to guess, I would say your wrist device had one purpose: to make sure your wrist was inside it.

Your phone app probably took care of any and all tracking required. That wrist device simply detected when you cut through that copper band, breaking the circuit, and alerted your phone via bluetooth. Your phone has all of the requisite tracking hardware, but it's not physically connected to you. Leave it behind, and you'd be out of quarantine!

...But not if they had a device attached to your body, which could detect if it's band had been disconnected.