It's been a while (a couple of years) since I last tried this, but I remember it being reasonably straightforward to get APKs of apps that were on the Google marketplace. Not through official user interfaces, of course. I can't remember if I ended up using some Chrome extension or a third-party app that needed root.
Not a particularly useful comment sorry, just "it was possible two years ago if you jumped through some hoops."
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.
App stores seem to be getting progressively more hostile to this kind of thing though - you can't just download an APK / iOS app, you have to do it through a device. This lets the stores do "app slimming" (and per-country / per-carrier customized apks) to remove resources you don't need (like binaries that don't match your architecture), which would change the checksum. Which is useful, but inconvenient for this goal.
Something like fdroid may be supportive of this, which would be cool. But I wouldn't expect the mainstream ones (or Apple) to ever embrace it - it'd be a bad user experience / wasted UI space in the vast majority of cases.