The connect chat idle timeout property - connect.support.idle.delay is triggered only when the user is connected to an agent using the URL for the queue.
i.e - The end-user should be connected to the agent where the URL is defined as below:
If the user is connected to a live agent using the virtual agent widget as below, the chat idle time will not be triggered.
This is an expected platform behavior.
Connect Chat & Service Portal are built on different architectures, and therefore the behavior will be different.
Also, if you inspect the virtual agent widget in the portal, the URL which is triggered in the background is pointing to /$sn-va-web-client-app.do?, which is not Connect Chat.
If you are using the virtual agent widget to connect to a live agent, you need to use the below properties: