Is there any reason why we can't offer a debit card to an authorized signer on an account?

It depends, in my opinion, on who requests the card. If the account owner authorizes the issuance of the card, I think it's fine. But the authorized signer should not get a card without the card owner's authorization (which may be part of the initial appointment of the authorized signer).

What is the person getting the debit card is not an authorized signer? And is a minor?

Would you give an unauthorized signer checks on the account? I agree with John. I would also check your account/card agreements to make sure that there are provisions making the account owner liable for all card transactions on their account made by the agent.

As to a minor, depends on your bank's best practice. Minors can't be held responsible for most contractual obligations (read the bottom of a card signature sales receipt) We do issue cards with limited access in some cases. (Minors over 16 and owner and signer of the account, parent must be joint owner and co-sign the card application)