Does clearing action tokens count as an action? That would mean that every time you cleared a character assigned this relic, you would heal them one click.

It doesn't, but free actions count. So if you perplexed or outwitted on your clearing turn, no heals. Leadership and poison area also free actions, and it's easy to forget at the beginning of the turn that using them will prevent the healing. That said, it's still a very worthwhile piece.

Couldn't have explained it better. Probability Control is about the only power they can actively use if they want to heal, as it doesn't require an action.