Regarding when the game can "detect" gamepads connected: This is a limitation of the Unity engine. Unity cannot detect newly connected devices while running in playmode (aka. running the game). Therefore you have to plug in your gamepad before launching the game for it to be detected and usable.
Seeing as you are trying to play the game on your Nvidia Shield Tablet (and I recon the game is streamed from your computer), it should still work in the same way as playing the game directly on your computer.
However, since the game is not currently using Exclusive Fullscreen mode, this might cause some issues while trying to stream to another device. We are currently testing using Exclusive Fullscreen, but do not have an ETA on when this might be added, as it has caused some issues in the past.
Hope this helps.