Agreed. From a privacy perspective, it is a lot safer to run the app in an environment where you have admin control. E.g. disable when not in use, block access to sensitive device information, limit background and network activity as much as possible.
I’d second this. The organization behind it (https://calyxinstitute.org/about) is a long established non profit, their portfolio covers various cool projects. The vpn client itself is basically a rebranded version of https://f-droid.org/packages/se.leap.bitmaskclient, it it is nothing special.
The question is not weather Google is tracking or not, the question is if Google is breaking the law doing so.
I love a good story driven Star Trek game. And it even has one of my favorite characters in it 🖖
If you want more in depth control on google services, CalyxOS supports microg in various combos (https://calyxos.org/docs/guide/microg/#options-for-running-microg-in-calyxos). While it has some limitations compared to the official google services, it also allows better privacy control and is fully open source.
This is your best option. I did do movie nights with friends during the pandemic in a similar way, but I used OBS Studio to create the stream and Monaserver to stream it to all users. I did not know VLC can handle the streaming to users directly, making this dead simple to setup without additional software. You just need to know how to configure your router to allow the needed port forwarding.