To understand this better, I would encourage you to read this document - https://docs.aws.amazon.com/lambda/latest/dg/invocation-async.html
Eventbridge is one of the lambda event sources that invokes lambda asynchronously, which implies that unless the lambda service itself is down, eventbridge will deliver the event in the lambda service's event queue.
If the lambda service cannot start a lambda instance because of throttling or any other reason, those events will stay in the event queue of the lambda service. As far as Eventbridge is concerned, it has already delivered the event to the lambda service.
What you need to do to handle your scenario is to also configure an "On Failure" lambda error destination DLQ, as explained in the same document.
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