in which scenario is a full proxy required?

A full proxy terminates the inbound client TCP connection and establishes a separate, independent outbound TCP connection to the server. This separation enables applying different TCP profiles, security policies, and traffic optimizations to each side independently, which is the defining purpose of a full proxy architecture.

Direct administrative access to servers is a management topology concern that requires jump hosts or out-of-band access, not a full proxy architecture.

The server sending responses through the load balancer describes a basic routed or half-proxy return path, which does not require full connection termination and re-origination.

When the server must see all original TCP client parameters, a pass-through or transparent mode is needed - a full proxy deliberately hides client TCP parameters by re-originating the connection.