I think the GP would have been fine forfeiting what he had paid for a cancellation. Homejoy was asking him to pay MORE for a canceled appointment than if the appointment had actually happened, ie he would lose what he had paid plus $10. How is that fair?
Homejoy provides a service. The OP cancelled that service. What service was rendered that required an additional payment to be fair to Homejoy?
Homejoy was certainly not fair to try and charge $10 MORE to cancel the order. But 'forfeiting' your credit card charge just because you could not or did not cancel in time, that doesn't skirt the boundaries of fairness on any service.
Credit card payment reversals keep merchants honest, they are disincentives to enact or continue unfair behavior against the consumer.