If VPC Flow Logs for the AWS Lambda ENI isn't showing any data going to the IP address of the ENI for AWS PrivateLink, then your traffic isn't going to the private link. The "NoData" status in your VP Flow Logs means no data was recorded during the five minute capture period that is set on the ENI. You can try inserting some extra debugging around the AWS Lambda code, which will make the function call to the logging API. The debugging code can also check the logging API configuration to make sure that it's configured to point to the AWS PrivateLink endpoint address.
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