I'm in the process of choosing a reliable private messenger ...
Can you explain what your requirements are? Because the choice of technology depends CRUCIALLY on what the technology is supposed to accomplish.
For example, just today I used Facebook Messenger, Google Chat, WhatsApp, cell phone SMS, plus of course cell phone calls. That's on a Sunday (so no work-related stuff), and we're travelling (so no home phone, nor walking over to chat with neighbors in person). Why do I use all these apps? Because of the old German saying: "Du muss die Leute da abholen, wo sie stehen" (you have to pick up people where they are waiting). If someone wants to use communications app X, I will adjust to it, within reason. Certain apps I won't discuss certain topics on (like discussing sex on my office's internal app would be a big mistake). Certain apps I don't consider secure enough for some exchanges.
... requiring a real phone number to register is a big red flag?
For certain uses, it is NOT a red flag. On the contrary, for those uses I consider it to be a real plus, and I inherently trust the app more, because I know that they have done a better job authenticating their users. For example, if I get a message on for example LinkedIn or Facebook from my friend "Adam Bob" with whom I have not chatted on that platform ever, I will initially be really careful, because those platforms do ZERO authentication of users when signing them up. On Facebook Messenger it is even worse, with the current wave of accounts being cloned: if I get a message from my friend "Adam Bob" who I have chatted with all the time, and suddenly their behavior becomes dangerous (like asking me for money because of some emergency), I usually find that it is not Adam Bob, but some scammer/hacker. In contrast, if I get a message via SMS from someone who has Adam Bob's phone number, or on Google Chat from someone who uses Adam Bob's gmail address, I'm initially more likely to trust the message (and when it gets to sending money, I would definitely first insist on seeing them in person).
I fully believe that there are other applications, where people see it as an advantage that no phone number is required for authentication. Great for them; for me, this is usually a disadvantage.