The visions of 5G and Beyond (B5G) imply unprecedented expectations toward high-performing connectivity services in both public and private networks. Connectivity services that offer performance guarantees along multiple Quality of Service (QoS) dimensions are partially available today, but are confined to (virtual) private network services. However, open and equal access to public and Internet-scale Specialized Connectivity Services (SCS) delivered on-demand with interconnections across networks and support for mixed traffic modes that go beyond traditional best effort does not exist. In this paper, we argue that this is a huge industrial and societal problem that needs a solution. However, this problem is highly complex and multifaceted, and there are many reasons why we are more or less locked into the status quo. The paper identifies the stumbling blocks and proposes a set of solution elements to take us across these hurdles, alongside related research topics. This includes an approach to “Multi-Level Best-Effort (MLBE)” and evolved net neutrality. Models and simulations are provided showing how a mixed traffic mode approach provides anticipated benefits. Arguments are given why the context brought by B5G will put us into conditions for change, allowing public SCS eventually at a global scale.