8.5.3 newmail popup feature requires 8.5.3 mail template for optimal function
If running with the new-in-8.5.3 newmail popup (which is enabled by enabling "Display a popup alert" in File - Preferences - Mail - Sending and Receiving), a change made to the 8.5.3 mail template will prevent saved docs from triggering newmail. Note that if you run with a pre-8.5.3 mail template, newly saved mail docs (whether replies, drafts, etc.) will trigger and be included in the newmail popup.
Resolving the problem
Use 8.5.3 mail template.