In which scenario is a full proxy TCP connection required?

TCP routing forwards TCP segments at the network layer without terminating the transport-layer connection, so no full proxy is needed - the same session flows end to end.

An IPv4-to-IPv6 gateway requires a full proxy because the client speaks IPv4 and the server speaks IPv6 - the proxy must fully terminate the IPv4 TCP session from the client, translate the addressing, and establish a new IPv6 TCP session to the server, since a single TCP connection cannot span two different network-layer protocols.

UDP routing is connectionless and does not establish TCP sessions at all, so a full proxy TCP connection is irrelevant to this scenario.

UDP to ICMP is not a standard proxied gateway scenario; ICMP is a control and error-reporting protocol and cannot carry application payload in a way that would require a full TCP proxy.