AFAIK, MMOs keep all the game state on the servers already. The difference is that what they send to the client is key deltas to the game state, which the client then renders. Stadia type services instead render that on the datacenter side and send the client images.
With their expertise at networking and so-on, Google might have been able to get a slight advantage in server-to-server communication, but it wouldn’t have enabled anything on a whole different scale, AFAIK.
IMO, their real advantage was that they could have dealt with platform switching in a seamless way. So, take an addictive turn-by-turn game like Civilization. Right now someone might play 20 turns before work, then commute in, think about it all day, then jump back in when they get home. With Stadia, they could have let you keep playing on your cell phone as you take the train into work. Play a few turns on a smoke break. Maybe play on a web browser on your work computer if it’s a slow day. Then play again on your commute home, then play on the TV at home, but if someone wanted to watch a show, you could either go up and play on a PC, or pull out your phone, or play on a laptop…