The Shared Game Timer can connect to many bluetooth-enabled console gamepads and use them as remote controls.
You first have to tell the Shared Game Timer what you want the buttons to do by creating a button action map. For example, you can set the 'X' button to 'End Turn'. This way, you can customize the gamepads any way you want!
Below is a list of gamepads and what devices they work on. The list is incomplete. Please send me feedback on gamepads you've tried, and I will update the list. All tests have been done on Chrome, unless otherwise specified.
If a controller works mainly on Windows/Mac, consider using a laptop, put it on a desk near the gaming table and put the Shared Game Timer into Presentation Mode.
8BitDo Zero 2
This tiny gamepad is the perfect size for a remote control! And it is cheap! I found it for $20 on Amazon. It has no rumble though.
Nintendo Switch Joycon
The Nintendo Switch Joycon controller is small and thus perfect for serving as a remote control, and it comes in two parts that can be given to different players!
I so wanted this to work, but it is a struggle to even connect to it, and then it turns out it does not support the Gamepad API. If you can make it work, do let me know!