In May we identified an additional flavor of the distributed cache problem introduced with January 2023 CU which was not addressed with May 2023 CU:
Application pools using a managed account that has been added after the farm was created might not be able to talk to the Distributed Cache server because these accounts are not automatically added as a member to the WSS_WPG security group on pure Distributed Cache servers where no WebApplications are provisioned.
With August 2023 CU for SharePoint Server 2016 we released a fix for this issue.
The fix for SharePoint Server 2019 and Subscription Edition was already released with July 2023 CU.