As per this query, I went into /sidekiq and lo and behold… there are 79K messages queued on Sidekiq, with more than half of them in the critical queue (and others in low). The default queue is empty.
Checking the critical queues indicate that most of them are Jobs::RunHeartbeat jobs. I’m quite sure that this is not normal, and should be why I’m seeing such weird behavior.
Then under the Busy tab I find that the process is listening only to default. It is not listening to low and critical for some reason.
MYSTERY SOLVED! 
Now the other mystery is, why it won’t listen to low and critical… Upon a VM restart it still only listens to default. More investigations in order…