There is no "router on Xbox 360." The Xbox 360 is a **client** — it connects *to* a router.
Some early 360 dashboards had "Internet Connection Sharing" (ICS) to let 360 act as a **wireless access point** for other devices. **Terrible idea** — weak radio, no security, no routing features. Never use.
Bottom Line: Buy a $20 TP-Link Archer A7 / C7. Wire the 360. Done.