If you want to have a tangential discussion about how you interpret Ring's terms to permit them to do wild things behind the user's back, that's fine; but it would have been better to be more clear about the tangential nature of your comments. If the terms allow them to do wild things behind the user's back, then they can do those things with or without introducing this feature. And they can also introduce this feature with or without the wild things; and with or without terms of service allowing those things. They're orthogonal issues.
In any case, you're mistaken about what the terms allow. When you paraphrased the terms as saying they can "hand over your footage to anyone so long as Ring believes it's 'reasonably necessary' to protect the rights or property of anyone", you neglected to account for the clause: "as required or permitted by law". Under the Stored Communications Act, 18 U.S. Code § 2702 (b), there is only a short and narrow list of circumstances under which it is permissible for a provider to disclose communications content without a warrant. The most pertinent is an emergency involving danger of death or serious physical injury (exigent circumstances), which is what the link in the OP regarding warrantless and consentless disclosures is about. But exigent circumstances are also a longstanding exception to fourth amendment search protections in general: law enforcement can break into your house without a warrant if there are exigent circumstances requiring them to do so.
In any case, you're mistaken about what the terms allow. When you paraphrased the terms as saying they can "hand over your footage to anyone so long as Ring believes it's 'reasonably necessary' to protect the rights or property of anyone", you neglected to account for the clause: "as required or permitted by law". Under the Stored Communications Act, 18 U.S. Code § 2702 (b), there is only a short and narrow list of circumstances under which it is permissible for a provider to disclose communications content without a warrant. The most pertinent is an emergency involving danger of death or serious physical injury (exigent circumstances), which is what the link in the OP regarding warrantless and consentless disclosures is about. But exigent circumstances are also a longstanding exception to fourth amendment search protections in general: law enforcement can break into your house without a warrant if there are exigent circumstances requiring them to do so.