Agreed that it seems like there is "something different" they need to be doing; either "instead of" or "in addition to" the current behavior. Given that part of GoDaddy's customer sell here is being able to have a turn-key phpBB installation as one of many different applications they support, a situation of "inexplicably and without kicking a failure back, some types of email notifications will fail by default" definitely doesn't seem like the right position for them to be holding.
Still waiting to hear back on the the still-open case with GoDaddy, but won't be updating that further here.
Thanks for all the input. Will definitely be considering what the config.php might be able to show me during future investigations.
-Trench