Initially I claimed that Azure private endpoints use CNAMES on the public DNS and that CNAMES are not supported, referring the following document:
https://learn.microsoft.com/en-us/azure/storage/files/storage-files-identity-auth-active-directory-enable#supported-scenarios-and-restrictions
However, I realised the documentation was referring to publicDNS (which was working), but CNAMES in Azure are supported. This contradiction was raised around 12/06/2023 and resulted in the first documentation being updated:
https://github.com/MicrosoftDocs/azure-docs/issues/110635
The only outstanding reference now to explain why this doesn't work with private DNS is this article:
Always mount Azure file shares using file.core.windows.net, even if you set up a private endpoint for your share. Using CNAME for file share mount isn't supported for identity-based authentication.
https://github.com/MicrosoftDocs/azure-docs/blob/main/includes/storage-files-aad-permissions-and-mounting.md