CW Log Shipping to Self Hosted Elasticsearch on EKS

EventBridge is used for events that indicate a change in your resources, for example - Describe*, Put*, Get* actions etc. And, for your use case 'Subscription filter' is an appropriate solution which is native to CloudWatch Logs. Using Subscription filter, which monitors for a particular pattern you mention - for any newly ingested events, you can set a destination (like a target in EventBridge) where these filtered out events are sent.

Please keep in mind that these filters don't work retroactively, and are meant for only the log events ingested once after the filter is created.

Read about subscription filters.