they had to burn them.

They say that, don't they?

Really I think the hassle is just worth less to the surgeon than to the patient so they're incentivized to lie about it.

With how many religions insist adherents keep their parts for later burial (or other reasons) a patient liaison has to have a waiver in a drawer somewhere you can sign (Formalin is toxic) so they can return your bits or they'd have to deal with church funded lawsuits or at minimum bad PR.