IMO it is no different than a payment plan set up by a vet. Only you pay some of the costs before certain treatments. They work out the average cost of shots, deworming, teeth cleaning and whatever else they do, and you pay in installments. If a person had incurred costs that they put on a payment plan, it would be paid out of their estate (and it is not unheard of for gyms in certain areas to ask that the remainder of a contract be paid out of an estate, I have seen it in the news before).
You sign a contract, with any contract comes terms, and a year's payments were one of those terms. If I financed a vehicle and crashed it with no insurance, I would still have to pay for the financing, without the vehicle. The contract at Banfield is the same thing.
Essentially, I agree with Denise, if you put all those charges on your credit card they would still be there, why not with Banfield's Wellness Program which is essentially a payment plan like a credit card.