This solution addresses two distinct requirements cost-effectively using a single Direct Connect connection. First, to provide on-premises access to VPCs in multiple AWS Regions, a Direct Connect gateway is the optimal solution. It is a global resource that allows a single Direct Connect connection to be associated with Virtual Private Gateways (VGWs) in any AWS Region (except China). This avoids the high cost and complexity of provisioning separate Direct Connect connections for each Region. Second, to provide on-premises access to AWS public services (like Amazon S3 or Amazon DynamoDB) over the private network connection, a public virtual interface (VIF) is required. A public VIF advertises AWS's public IP prefixes over the Direct Connect connection, routing traffic to these services through the private link instead of the public internet.

B: Setting up additional Direct Connect connections is significantly more expensive than using a Direct Connect gateway and is not the least-cost solution. C: Establishing a Site-to-Site VPN over a private VIF is not a standard or valid architecture for extending connectivity to other VPCs in this context. D: Using a VPN over a public VIF to connect to VPCs is an unnecessarily complex and less efficient method compared to a Direct Connect gateway. F: While VPC peering connects the VPCs, routing on-premises traffic through one VPC to reach others is less efficient and scalable than using a Direct Connect gateway.

1. AWS Documentation - Direct Connect gateways: "A Direct Connect gateway is a globally available resource. You can create the Direct Connect gateway in any Region and access it from all other Regions. You can associate the Direct Connect gateway with a virtual private gateway for a VPC. You can then create a private virtual interface for your Direct Connect connection to the Direct Connect gateway." This supports using a DX gateway for multi-region VPC access. (Source: AWS Documentation, Direct Connect User Guide, "Direct Connect gateways").

2. AWS Documentation - Direct Connect virtual interfaces: "Public virtual interface: A public virtual interface enables you to connect to all AWS public IP addresses." This supports using a public VIF for accessing AWS public services. (Source: AWS Documentation, Direct Connect User Guide, "Direct Connect virtual interfaces").

3. AWS Documentation - Working with Direct Connect gateways: "After you create the Direct Connect gateway, you can associate it with the virtual private gateways for your VPCs... A Direct Connect gateway supports communication between its attached private virtual interfaces and associated virtual private gateways." This confirms the architecture described in option A. (Source: AWS Documentation, Direct Connect User Guide, "Working with Direct Connect gateways").