It seems that the problem may come from the fact that the xbox 360 still establishes a connection using the built-in wireless card even when if connected via a wire to the bridge (if you configured the xbox for your wireless network at one point)
Xbox 360 controller wireless receiver driver update doesn Jul 14, 2020 Creating an Internet Bridge Connection for XBoxLive : 3 Open Your Network Connections. Skip this step if you already know how to get to your Network … The Bridge for Xbox 360 Reviews - Metacritic The Bridge for Xbox 360 game reviews & Metacritic score: The Bridge is a 2D logic puzzle game that forces the player to reevaluate their preconceptions of physics and … Control your Raspberry Pi by using a wireless Xbox 360
How to Bridge a network connection from a - Xbox 360 Have an Xbox or Xbox 360 that you want to connect to the Internet and a notebook PC? With this gamer's guide, you'll learn how to connect an Xbox to Xbox Live via your laptop's wireless Internet connection — foregoing expensive wireless adapters. For complete instructions, watch this free video tutorial. Wireless Ethernet Bridge for xbox 360 - Wireless Apr 09, 2006
Then you can connect the Microsoft Wireless Receiver to Steam Link and do the connect between the the receiver and your wireless 360 controller. So yes if you do not have it, you need Microsoft Xbox360 USB Wireless Receiver.
Jun 17, 2020 Amazon.com: HDE High Speed USB Wireless WiFi Network