Turns out the reason was that too many processes were competing for the CPUs; the server tried to do too many things at once and in the end it did none of them well.

The server was constantly switching between tasks and none of them could make use of the CPUs fully.

I moved RabbitMq out, onto a separate machine and both the Rabbit and other services that remained on the original machine now perform as expected, CPU usage went up on both machines and the responsiveness just as well.