No. A thousand times, No.
The simplest analysis is that the check is not payable to him, it is payable to the trust.
He can certainly affix an endorsement in the name of the trust. That would make it a bearer instrument. If you were going to cash it for the individual, then you would require him to sign it again with his own name, showing that it was him, not the trust that received the proceeds.
That paper trail would indicate that your bank was very clearly aware of the likelihood that the trustee had breached his fiduciary duties. If the funds had indeed been diverted from their rightful owner, the trust, your bank could and should be found liable for facilitating the breach.
It's frequently true that people set up inter vivos trusts naming themselves as trustees with no comprehension whatsoever of what they have done. The trustee may think you are simply a nitpicker, but what you are doing by requiring that checks payable to a fiduciary be deposited is carrying out the trustor's stated intent, whether he or she understood it or not.