Privacy is really the wrong word in my opinion. What you mean is confidentiality. This might sound like nitpicking, but private communication might mean that so

Privacy is really the wrong word in my opinion. What you mean is confidentiality. This might sound like nitpicking, but private communication might mean that some attacker does not know who is talking to whom, which is not what TLS even tries to provide (anonymous communication). The term is somewhat overloaded. You can have fully encrypted communications and still give away information you intended to be private. TLS does try to provide confidentiality and part of that is authenticating who you are talking to. Without this your communication is not at all confidential, as anyone could impersonate your intended target. A passive adversary can't read your traffic, true, but an active attacker (of for example the coffee shop dwelling variety) is not a major step up and would undermine the confidentiality aspect entirely.