Here is the setup I did
Expected Result: The quote created in 4 would be routed to the user as an Omnichannel work request.
Actual Result: The work request queue for this user remains empty.
So how do I get something routed using this queue?
So I'll answer my own question. The missing element was a Presence Status linked to the Service Channel. So summarizing what is needed, to save someone's future agony, you need the following:
You do NOT need a flow; just changing ownership of the routed record to the queue will route the item.