Is a check endorsement required if the check is being deposited into an account in the name of the person the check is written out to?

Yes, but typically it may be done by the bank with some form of an endorsement guarantee stamp.

Wouldn't the customer writing "For Deposit Only" on the
back of the check be enough?

That's not enough. An indorsement is defined as including the signature of the holder. That signature can be in ink or pencil, typewritten or pre-printed. It can be a thumbprint or anything else that has been adopted by the holder as his/her/its signature and accepted by the transferee as a signature.

There are times that a drawer really wants the manual signature of the payee. Insurance companies, for instance, often require a manual signature as demonstrating that the payee was alive when the check was transferred.

A bank, as suggested above, is in a rather unique position of being able to supply its depositor's indorsement if necessary. But that's a "crutch" that a lot of people use as an excuse not to require customers to indorse (with a signature). If the customer's signature appears in the indorsement (rather than the bank's rubber stamp), it's easier to demonstrate that the payee received value.

My understanding is it depends on if you're the depositing bank or paying bank. Your tellers should be sure that there is an endorsement or, if depositing to the payee's account and there isn't, stamp it "AEG". If you are the paying bank and there is no payee endorsement other than the bank of deposit and that bank is located in a state that has adopted the revised articles of the UCC the banks endorsement is enough. New York has not accepted the revised articles.

I am not up to speed on a revised UCC that would allow a "For Deposit Only" endorsement. It would however seem to me that one should read the terms printed on the signature cards which is the contract the bank has with its customer. Usually their is some verbiage indicating the bank agrees to only allow transactions conducted by the signers on the account. If you allow just a "Deposit Only" you have no way to prove compliance with that provision of your contract with the customer.

While it dosen't happen often accepting a deposit of an item made payable to your customer by someone other than the customer using a "Deposit Only" and no signature endorsement can cause a loss to the bank.

John is absolutely correct.