They do indeed care.
My brother and I submitted a sample from my dying mother, which they were unable to process properly. We were offered a replacement kit, but at that point our mother had already passed away. When I conveyed this to 23andme, they refunded our kit cost and removed our mother's pending profile from my 23andme account.
From their email:
"Unfortunately it is not possible for us to unregister a kit after an initial sample fails analysis. Replacement kits are automatically registered to the individual the original kit was registered to, and they retain personal registration information including name, birthdate, and legal consent to our Terms of Service and research document.
If we allow someone else to use the replacement kit, the sample would not be associated with their personal information. Additionally that person would not agree to our Terms of Service and provide legal consent for us to process their sample. For privacy and legal reasons, a replacement kit cannot be used by another individual."