The search for a "YouTube Android 6 APK" is the last gasp of a dying ecosystem. As Google moves further away from legacy support, the friction of maintaining these apps on older operating systems increases.
Users factory resetting Android 6 devices today are often met with the dreaded "Your device isn't compatible with this version" message, effectively bricking the device for YouTube use via the official store route.
Since you are not using the Google Play Store, you must allow your phone to install APKs manually. youtube android 6 apk
Not all APKs work on every Android 6 phone. You need the right architecture.
Google Chrome on Android 6.0 is surprisingly robust. The search for a "YouTube Android 6 APK"
Before side-loading any APK on Android 6.0, go to Settings > Security and enable "Unknown Sources." Remember to disable this setting again immediately after installation to keep your device secure.
For a standard, safe, working YouTube experience on Android 6: Since you are not using the Google Play
If you absolutely must run the native YouTube app on Android 6.0, you have to step outside the Google Play Store. This involves side-loading an APK (Android Package Kit). However, this comes with significant caveats regarding security and functionality.