If it does work fine without Google Play service, then there’s not much you can do with this kind of pop up other than modifying the APK (or perhaps using some kind of hook framework like Xposed).

You could try something like microG, but your ROM would need to support signature spoofing.

btop is not Bazzite specific, it’s just preinstalled on Bazzite. You can install it on pretty much any distro.

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Seems like the website you’re trying to access is using HSTS, which means it has told the browser “if the SSL is invalid, do not proceed”.

If the same issue occurs across multiple devices, I would be inclined to think that there’s some problem with the website.