Ansible tmp files created with odd permissions
The reason for the failure is that the server in question, unlike all the others, had sftp disabled. 
Why the error messages are what they are, I don't know (why the errant scp transfer mechanism failed?), but I don't have time right now to investigate with the debug option enabled.

But that was the issue, and adding scp_if_ssh=True to the [ssh_connection] section of my ansible.cfg solved the issue:

[ssh_connection]                                                                   
scp_if_ssh=True