You can probably still get the one that'll install on your device(s), but if there's customization for e.g. carrier X in country Y (or app slimming) you're unlikely to know or be able to find it from the infinite "download APKs free now!" sites.
And the last time I looked, all the apk-downloaders required your device ID, because Google's API does (for customization reasons) - it's much more of a "you can do this if you emulate a device" than "you can download it". I'm also not sure if the ID works unless you have gplay installed, which you may not have if you're being careful/paranoid enough about security to manually validate apps.
Not a particularly useful comment sorry, just "it was possible two years ago if you jumped through some hoops."