In the UK at least, the vendor needs to show that their process was transparent and fair in order to fulfill their side of the contract. One of the examples specifically called out as unfair is "renewing without sufficient notice".
In this case, they didn't send any kind of email before or after payment.
For small enough amounts, at least in the US, the credit card company will sometimes just accept it and give you a courtesy credit without going through the whole investigation.
And regardless, having too many chargebacks will often get the merchant into trouble. Just having to deal with the hassle of it is bad for their business, a good way to get back at them for poor service.
In this case, they didn't send any kind of email before or after payment.