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Google gets just as much data as it would get if you would use sandboxed Play Services
Oh really?
Aurora allows you to spoof your device model, along with using anonymous accounts, used by many other users at the same time, which helps you blend in, now assuming we also use a VPN, and you are Google free except for Aurora, it makes it way more difficult to track your usage back to you compared to using a broader amount of services in the Play Services ecosystem, linked to an individual account.
And that's only the privacy side, but if we go to the fundamentals, I assume that Aurora only exchanges the minimum data necessary for the app to work, where for the Google Play Store, it's likely that it exchanges quite a bit more data than what we could consider the minimum to download an app (although that is just an assumption of mine).
Don't get me wrong, I am not a developer, but I find it a bit of a bold claim to say that Google gets as much data about you if you use an anonymous account in Aurora, compared to using the Play Services along with the Play Store, I would be glad to hear how that works.
Security wise only, yes, the Play Store is a safer way to get your apps.