How can the process of allowing the user to respond to an DNT request with an MDN be made less painful and less likely to upset privacy advocates or at least prevent people from accidentally confirming their addresses to spammers?

Well, let's consider how traditional mass (snail) mailings determine whether the recipient has read their mailing:

The envelope contains two items of consequence:

1. the letter/message containing whatever important body text the sender has.
2. a postcard response card to the sender (which will be handled by the reader after reading the letter), if the sender really wants the response, the postcard may have a marker which causes the respondent not to need to pay for or apply postage.

How could this approach be mirrored in a mail client?

Well, for one the MDN prompt (again, it shouldn't be a dialog) could be placed at the bottom of the email.

So how should the DNT be indicated?

Supposing Mail wanted an inline prompt (again anything is better than a modal dialog), and there is definitely merit to a more visible indication than the simple status icon, the bottom of the email could show another message part.

===================================================
.___. Q <Que@example.com> asked for a return
|...| receipt from Jane Doe <jane.doe@example.com>. 
|.. |
|x__| You may take the following actions:

 
 No
 Undeliverable   
 Received        
 Notice 

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