A payment processor isn't (at least where I am) entrusted with any special powers to make judgement calls on whether you deserve your money back based on a potentially existing contract between you and a third party - that's the court's job. Cooperating with them makes it easier, speeds up the process and ensures an ongoing relationship with them, but you don't have to prove that your card was stolen (which is hard to do anyhow) - they have to prove that it was you that authorized the charge. Getting your money back without their cooperation may involve taking them to court though.
There's a similar thing with SEPA direct debit. You can pull money from any account (it's trust-based; you're required to provide documentation showing that you were authorized to do so when asked), but the account holder can pull it back for a certain amount of time (iirc, it's six weeks, at least here). The banks do not act as a judge here, they simply put your money back into your account and inform the other bank that the charge has been reversed who then in turn takes it out of the pulling account. If the other account holder believes the charge back to be unlawful (i.e. fraud, or or a charge back because of insufficient balance), they have to bring legal action, but they can't use the bank as a tool in the process.