In layman's terms, you, as the depostory bank, warrant that the person who transfered this check to you (your customer) was the proper person to do so (among other things, but this is the portion that is relevant to this discussion), and that you have no knowledge of any facts to the contrary. You're making this warranty whether the prior endorsement is present on the check.
Since a check payable to "cash" is a bearer instrument (it can be negotiated by anyone), it doesn't need to be endorsed by a specific person, like a check with a specific payee does. But the endorsement still has value as it relates to memorializing the transfer from your customer to you. I'd suggest following through with your orioginal inclination and requiring the customer to endorse the check prior to acceptin gthe item for deposit.