When deploying live chat buttons with routing type of "Omni-Channel Skills", Salesforce returns an error "Required fields are missing: [RoutingType]".

This appears to be a known issue with Salesforce Change Set / API for chat button deployments. As a work-around, you can update the routing type field to another value (ie LeastActive ...), deploy and modify the routing type back to "Omni-Channel Skills" in the target org.
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Note that this error only applies to NEW chat button deployments (or the first time you change the routing type of an existing chat button in the target org). After the routing type is updated in both source and target, subsequent deployments will no longer error.