VirtualBox's Proxy Preference Does Not Configure Guest Applications
Adapter mode supplies a network path; the guest browser or package manager still needs its own supported forward-proxy input.
VirtualBox's HTTP proxy preference configures VirtualBox itself. It does not write matching settings into the browsers, package managers or updaters inside its guests.
NAT and bridged networking likewise change network attachment, not HTTP client behavior. Neither turns a normal socket connection into an HTTP CONNECT exchange.
Separate the diagnosis into two questions. Can the guest reach the required listener through its virtual adapter? If so, does the affected guest client select that listener's type, address and account?
If no client can reach the host/port, investigate connectivity first. If the browser works but package downloads fail, inspect that package client's supported settings. For a guest using APT, configure APT’s own proxy options inside that guest.
Bridging may change reachability and the guest's presence on the physical network. Choose it for that requirement rather than as a substitute for setting the guest application's route.