so can someone explain why a pretty robust dedicated device is necessary as a replacement?
The cloud is just someone else’s computer, so when you cut the cord from the cloud, you gotta run your own server.
And you don’t need to buy a (robust) device to run HA, just install it on a spare system and start playing with it. I started building mine about 1.5yrs ago when I bought a house and I think I only gave mine like 2 CPU and 8gb ram.
What actually happens when I turn on a smart switch in my home? Does that command have to be sent to a server somewhere to be processed?
Yes, you have to have something that accepts your commands and sends the action to the end device. Just like your Google home did.
What really has to be processed, and why can’t a smartphone app do it?
Because that’s not how things work. Your app has to talk to a server to send the commands, Google home has cloud servers and a local bridge. HA has an app that you can use to control your stuff, same as Google Home.
Smart Home apps are worthless without hardware required to connect the app to your home.