TL;DR: Android has a secret which, apparently, is still not well known. It is... the 'skip' button, because you do not need a Google account to use your Android smartphone.
Recently my wife switched over to a new mobile phone ( Fairphone ). During the setup, I showed her the 'skip' button. With that, you can skip the Google registration of your phone (which most people believe is mandatory, but is actually optional). So you are not required to agree to Google's Terms of Service.
After skipping the registration and entering Google account credentials, you will be presented with an Android phone. The things you'll likely notice to be 'missing' is the Google Play Store, Google's Youtube etc... After installing F-droid, an open source marketplace, you'll be able to find and install many useful applications. If you still need some proprietary apps, most of them are also available as an APK download from the website of the vendor (example: Signal and Whatsapp).
If a vendor only provides their app via the Google Play Store, contact them about it. Of course you can download the APK via a 3rd party website, but that can be insecure. It would be much better if the vendor would publish the APK (or even better, make the application open source).