As a manager - I've had to do crap like this now and then to light a fire under people. It can be done to indicate extreme urgency. There also are real-world implications to some delays - for example not being able to run payroll is a legal nightmare and an ethical nightmare for hurting your employees. Generally, when I've had to do this, it's been my fault for not manangeing properly.
I'm curious, does it actually lead to faster resolution or does it lead to engineers spending time thinking about how to deliver the bad news versus actually focusing on the problem?
Usually this request, though frequently very rare takes the form of "Issue $ISSUE is causing critical impacts on $IMPORTANT_THING, if we don't solve it by $TIME, it will cause $BAD_IMPACTS. Please make this your top priority and redirect questions about all other items to me. If you think this will not be done by $TIME, please let me know as soon as possible. I appreciate your hard work" Then you tell rest of the org: "$TEAM is working on $ISSUE and is re prioritizing other commitments. If you have any questions or concerns, please reach out directly to me."
If the developer turns around and says "this won't happen in time", that's a problem for the manager, not the developer in like 99% of situations. It's not "bad news", it's the truth, and you should always be able to tell the unfiltered truth (especially to your direct manager!) without any repercussions.