Steam’s Web Pages and Game Traffic Need Different Network Paths
Check the failing Steam component and its TCP or UDP requirements instead of treating a working browser proxy as support for every game connection.
A browser can open the Steam store through an HTTP proxy while the Steam client or a game still cannot connect. Those operations do not all use the browser's HTTP transport.
Valve lists both TCP and UDP requirements for Steam operation, with additional UDP traffic for gameplay, peer-to-peer networking, voice and Remote Play. Start with the specific component that fails: web content, client connection, game session or local streaming.
An ordinary HTTP CONNECT proxy gives a TCP tunnel when its policy permits the destination port. OpenSSH's dynamic SOCKS forward also provides TCP forwarding. Neither setup automatically supplies the UDP traffic needed by a game.
SOCKS5 defines UDP ASSOCIATE, but using it requires support in both the redirecting client and the proxy server. A product accepting a SOCKS5 address for TCP does not thereby implement UDP relay.
Check the current game's networking requirements and the organization's permitted route. Changing one Steam client transport setting does not rewrite the network code of every game launched from it. Valve also notes that its required ports cannot simply be remapped to HTTP or an arbitrary custom range.