The recommended service for establishing a private, high-bandwidth, and reliable connection between an on-premises data center and the cloud provider's network, without using the public internet, is AWS Direct Connect. This service satisfies compliance requirements that necessitate bypassing the internet and provides the necessary bandwidth and consistency for sensitive workloads. Other options like VPN use encrypted connections over the public internet, which might not meet certain compliance requirements. VPC Peering is used to connect two VPCs within the cloud provider ecosystem and is not applicable for on-premises to cloud connections. SSL Encryption over the public internet wouldn't provide a direct interface with the provider's network and wouldn't fulfill the requirement for bypassing the internet.