Interesting tidbit: I've been watching "the big bang theory" a lot these past few weeks on my own hosted jellyfin install.
I don't use google search anywhere, I don't type tbbt anywhere. Yet, on my Android phone I have this obligatory Google news thing when I swipe left (HATE that) and all of the sudden that thing got chock full of chatgpt written TBBT articles... I don't really go there (usually end up there by accident swiping left once too many) and I don't read those articles but it really obviously switched to TBBT articles when I switched to watching TBBT.
This really kinda freaks me out and makes me wonder WTF more google is monitoring. I use a Google Chromecast, I guess google monitors that?
Nope, google home assistant thingy still listens and saves things you talk about. My wife and I occasionally have conversations, and the next time she uses her android phone, it shows her ads for whatever it ismwenwere talking about. When this happens, her phone isn't in the room, though I'm sure that may also listen.
The smart speakers are a privacy nightmare, but on your phone, random apps cannot access your microphone without your knowledge or control, excluding straight up malware. There's a million anecdotes like yours on social media and none of them have held up to rigorous testing.
The fact is, most people don't realize how much of their thoughts one way or another are reflected in their digital lives, nor how good google's algorithms are. And there's also things like the frequency illusion etc.
Unlike other ROMs like CalyxOS, GrapheneOS ships the actual Google Play services and Google Services Framework, but they're sandboxed so they can't spy on you. Whether your banking app works, depends on the SafetyNet requirement it has. If it requires full integrity, it unfortunately won't work on any system that is not whitelisted by Google. If basic SafetyNet integrity is enough for the app, it will work on GrapheneOS.
You can google (or duckduckgo) your banks name and "GrapheneOS", and you may find posts by other people who tried it out.
Edit regarding software in general: When you install Sandboxed Play Services, you also get the Play Store which you can use to install apps. You can also use the Aurora Store, an anonymous front-end for the Play Store. There is also F-Droid, which offers FOSS alternatives to most applications.
Edit 2: I found this post over on PrivSec which has a list of apps and details on whether they work
Which phone do you have? I've been able to disable that "swipe left to access Google" thing on every Google Pixel I've ever owned. Just long-press your home screen and go to home settings and disable it.
OnePlus 8T. One day they pushed an update that removed the option to disable it. Obviously they have a contract with Google so that they profit from this, and it sucks .
When I have some time I might reinstall some new OS on it that removes all that crap