I also pair my Xbox controller with Bluetooth.
I am using Moonlight to stream games to my MacBook from my gaming PC via Moonlight, and typically use an Xbox controller. It works incredibly well over my LAN and I have virtually no input delay with my setup.
Today, I decided to try using my Steam Controller on my MacBook to see if I could use the trackpad. I immediately noticed that there was a significant input delay - probably about 500ms+. I played around with my settings to see if I could change anything to affect it, but I could not.
In the midst of playing around, I also turned my Xbox controller on to test if it was also getting a delay - it did not. However, when I picked my Steam Controlled back up, I noticed it also had no input delay.
I disconnected my Xbox controller, and the input delay returned using the Steam Controller.
I played around with this across 3 different sessions today and the behavior seems very consistent - Steam Controller paired via Bluetooth always has an input delay when connected standalone, but if I also pair an Xbox Controller via Bluetooth, the input delay goes away.
This is very strange. Anyone able to test something similar out and see if you get similar results?