I'm not saying a new mobile OS alone would magically remove the Play Services requirement. But rejecting the duopoly means rejecting schemes like this reCAPTCHA requirement as well.
Yes, I know Android is Linux. But let's be frank, you're a business. You have a direct interest in keeping people on Android. If everyone abandoned Android for something else, your business model would be threatened. So of course you're going to defend the status quo.