Xbox 360 Elite Controller on Original Xbox (Arcade)? Not Compatible
No. Xbox 360 Elite (or any 360) controllers do not work on the original Xbox (often called "Xbox Arcade" or "OG Xbox").
Why
Protocol: OG Xbox uses a proprietary USB-like protocol (different packet structure, 1 ms polling). 360 controllers use a custom 2.4 GHz RF or wired USB-HID with different descriptors.
Port Shape: OG Xbox ports are oversized proprietary; 360 plugs are standard USB-A.
Wireless: 360 wireless receiver requires 360 console firmware; OG Xbox has no pairing stack.
What Works on OG Xbox
Original "Duke" or "Controller S" (wired).
Licensed third-party OG Xbox pads (Mad Catz, Logitech).
Adapter: Brook "X One" or "Wingman XB" adapters let you use modern Xbox/PS4/PS5 controllers on OG Xbox via USB.
If you're playing on a real OG Xbox, grab a Duke or Controller S—they're cheap on eBay and feel authentic.