There are third-party clients for the Play Store (Aurora store being a good example). Aurora store uses anonymous accounts to download the APKs directly from Google. That being said, just because you can install the application doesn't mean it will actually work without Play Services installed. I've had quite a bit of luck with random applications I've installed (interestingly most Google apps like Gboard, Photos and GCam work fine offline and without Play Services), however YMMV.
Aurora store does NOT let you download paid apps. If you have paid for a app, you can sign into that account in aurora store and download the app you bought. However, the paid app will most likely not work as most apps use a SDK provided by google for verifying purchases in a app. This SDK heavily relies on google play services. And secondly, using a 3rd party store like aurora does violate googles use agreement which means google could ban your account if you sign into it from aurora. I would highly advise to not use a google account you care about with aurora.
Nah, Aurora only works for snagging free apps from the play store via a proxy account- you're thinking of another well-known APK download site starting with an A, one which allows users to create their own 'app stores' (ie. repositories) and is rampant with piracy. I'm sure it comes in handy for kids with more tech--savy-ness (enough to avoid the malware!) than literal cents.
> you're thinking of another well-known APK download site starting with an A
Aptoide. I have seen pirated paid apps on Aptoide, but any app marked as "verified" is not pirated (as in, it's available free of charge elsewhere) and the app's signature on Aptoide matches the app's signature on Google Play. Everything in the main "apps" repository and some apps in other repos are verified.
Aptoide is useful for downloading older versions of Android apps, especially when APKMirror doesn't have an entry for the app.
Fun fact: Aptoide is open source and F-Droid is actually a fork of Aptoide.
