We cannot use Automated Invitations with the Embedded Chat component in our Experience Cloud site. This is a limitation in Salesforce.

Reference Article:
https://developer.salesforce.com/docs/atlas.en-us.238.0.snapins_web_dev.meta/snapins_web_dev/snapins_web_customize_chat_invitation.htm

Vote for this idea:
https://ideas.salesforce.com/s/idea/a0B8W00000GdbR2UAJ/automated-chat-invitation-support-for-live-agent-snapins