I do not know how Google home works so I cannot speak for it. I'm assuming with technology it will control your Google amplifier and Google speakers.
consider this. By the time you spend two or $300 on an amplifier and receivers that can run Google home and connect to the internet and all this other stuff, what do you think was left for the actual sound reproduction and/or sound quality. Too much of a good thing might be a bad thing. Yeah I am sure for 2 or $300 you might be able to find something similar but what kind of quality could you really expect. Also me personally I find hitting the volume on the remote or turning a knob, much easier than farting around with my phone to control everything. An old receiver can be made Bluetooth simply by a cheap $25 Belkin dongle. So as people are dumping amazing quality receivers to get Bluetooth, I am waiting in the wings like a bad magician.
Your TV unless it has an amplifier built in will not control volume or speakers. It's sends audio out signal but without power to run speakers. If your TV does send power that's pretty cool. My old Pioneer big screen had a kick a.. amp in the bottom. You can however run something like that with speakers that have built-in power like computer speakers. And they are a decent option. I actually kind of thought this might be a good option for you. It will require no amp. The speakers have the power built in. And for two or $300 you can get a decent sounding pair. I don't think you would sound any better than my $60 B&W, Goodwill amp, set up but it has some nice benefits. You don't need an amp and there's less chords. The downfall is you can't hook a lot of things up to it. But if you're only going to use the stereo to watch TV and or listen to music through Chromecast and put it in the TV you can use computer speakers or a soundbar.
That quality is going to be far less than the worst of worst systems. You will be able to find something new for 2 or $300 that has speakers and all the things you want but honestly how could all that be packed into one with any kind of quality.
For the other route you simply need an amp and some speakers. Don't worry about wattage or any of that for now because you will be looking at spending $250 on some used speakers which is a decent budget and $50 for a used amp, which can be had easily through Craigslist or Goodwill. You will need my help and I will gladly help you setting up a simple amp with some speakers. Some of it depends on what output your TV has. I actually had a TV that only has optical digital out. I had to buy an optical to RCA adapter for the amp I had at the time. But if your TV has RCA audio out than any little amp will do for now and you'll spend the budget on the best speakers you can find. Tell me where you live and I'll check your Craigslist ads and help advise you if you want to get a nice pair used. As for the sound bar option or computer speaker option you just buy a computer speaker and hook it up to your TV and you're good to go. I'm just kind of scratching the surface along here feel free to ask any questions I don't mind helping. Trying to keep this simple. Either an amp and speakers or powered speakers basically are the main options. Here's a pic of some powered speakers . I've heard some of these sound good.
Last a gentleman did tell me he bought some Bluetooth speakers that he said has his whole house wired with and they're all connected to the internet and Bluetooth. They were called sonos and i think they're what you were talking about. He said they sounded good and they were like $250.