Yeah, I've got 4 Stadia controllers left from the service. Between the free hardware and getting all my game purchases refunded I made out really good.
I only have my founders and a white controller, but besides that I did make a profit on Stadia! I sold the Chromecasts that came with the Founders and première kit. Also, I got the premiere kit free with Cyberpunk.
Only thing that wasn't a direct profit was the founders entry, but that was a birthday gift to me.
Although I have a great pc now, I still miss Stadia...
Exclusive to the OLED, it has a different wireless chip. From my rough understanding, it supports BLE which lets it stay on while the deck is asleep. Controllers that also support BLE can connect to it and wake the deck up that way.
For LCD decks, I've heard of people buying a smart plug and using that to toggle the Dock charger off/on. If you have wake on charge turned on in bios, this will let you remotely wake up the LCD deck from your phone or a home voice assistant.