Ops team?! We, the developers, are monitoring the DLQs: It is our mistakes, and we must fix them. Operations' only role is to keep the VMs the ActiveMQ is running on active, and the database it uses responsive.
If you used it in a synchronous fashion for a end-user, you are correct: He'll get a timeout - but the message will blink red on the DLQ, giving the devs exact info as to what has happened. As opposed to a 500 or something similarly bad, which probably in the best case was logged somewhere, and hopefully someone is tallying up the error codes every few weeks..
I fail to see how messaging is worse in this case.
it immediately demands the attention of the ops team that you've thrown all problems at while hoping that the magic queue never stops being magic.
while the client is sitting there going "WTF, where's my shit?"