I have set up Live Chat buttons with Omni-Channel Queues for routing. A custom visualforce page was created to handle routing using buttons, this is working fine.
I should mention I am using the code snippets from Service Cloud Chat Developer Guide instead of the Embedded Service for Web Developer Guide.
In the Lightning Service Console when the Agent goes "Online" the "Let's chat" button comes on. The routing and everything else works as expected.
On the other hand when accessing the Classic Service Console, when the agent goes "Online" the live chat button displays Offline as if there was no agents available.
I came across this article which states that for Omni-Channel routing "Agents can chat with customers in the console in either Salesforce Classic or Lightning Experience."