NVC has its flaws. It won't always work. Nor will anything else we know of. If something comes along that works better (higher "success rate"), I'll jump on that bandwagon.
If you can't be bothered to read the rest of this long comment, here's a TLDR: Most communications books, written independently, have the same elements as NVC. The other books are better at explaining why these elements is important. But the NVC book is better at providing recipes, which is why it became more popular.
A few years ago I did a deep dive on the topic, and read 4 books on communications: Difficult Conversations, NVC, Crucial Conversations and Getting Past No. Most of these were written/invented independently of one another. When I had read all 4, I looked over all my notes, and noticed that they all point to the same things, with only minor differences:
1. State your observation, without judgement, and be specific, not general.
2. Personalize and state its impact on you. The "feelings" part of NVC - all the books explicitly call out that you should explicitly state your feelings. The Difficult Conversations authors pointed out that many people in professional settings have the notion that discussions should be absent of emotions ("objective). Yet if a discussion at work is getting heated, emotions are clearly playing a big role. Don't assume you can solve the problem without bringing those emotions to the table. And indeed, many solutions fail because they failed to address the emotions which resulted in a lack of commitment. As one book put it: Unvalidated emotions are a minefield.
Just yesterday there was a nasty discussion at work about whether to split a monorepo into smaller individual repos. One side's work is bogged down by it being a monorepo, and the other side would have trouble if it were split into monorepos. Both sides discussed the their problems, yet neither side acknowledged the other's pain. Validating the other's emotions is a core principle of all communications books I've read.
(The issue did not get resolved by the end of the day).
And personalize: Don't talk in vague principles, but talk about how you are affected. This is where a lot of the tech world fails. I recall one heated discussion at work where a person was strongly pushing for something, and using all kinds of principles/analogies. He wanted someone else to change his workflow because it was causing problems for him. Yet, he never explained how it was causing him problems! When queried, he kept invoking analogies and principles. The world rarely yields neatly to principles, and every analogy has a flaw. You'll often get competing principles. Invoking them alone will not solve the problem.
The other element of stating the impact on you is discussing the needs. What need of yours is not being met? Is this impacting your efficiency? Are you wondering whether the work is futile, and not giving your purpose? Do you think you are being ignored and want consideration?
I cannot stress this enough: If you have trouble putting a "need" to your problem, it is because you haven't thought this through, and it's a bad idea to go into a conversation about the issue. Even if you don't explicitly state your need, this is a valuable exercise to do internally. Feelings are easy: When you're mad you know you're upset. Most people stop there and go have an argument. NVC forces you to start introspecting: My teammate did X and now I'm mad. But why is his doing X bothering me? Yes, my teammate undid all my changes and replaced it with crappy code. But why exactly is this bothering me? Your needs and his needs may conflict. But unless you can determine both, the conversation is likely going to go south.
All the communications books are part "therapy": Emphasizing introspection along with communication.
3. Request (optional - depends on the situation).
Regarding the "canned robotic phrases": This is akin to any other book that becomes dogma (think TDD, unit tests, referential transparency, etc). And just like the others, the original author does not insist you use these phrases. The NVC author explicitly says that the elements must be present, but the order doesn't matter, and sometimes it can even be communicated in one phrase. It nevertheless provides those templates to make it easy on beginners.
When I read the NVC book, I felt similar to you: These phrases are very artificial. And would annoy others. NVC is the only book that has these templates, and it is likely why it is very successful. The other books discuss the issues involved in conversations much better than NVC does - but they don't provide a template.
Now how artificial are these? Surprisingly not very. All my skepticism went away once I started observing people destressing a situation at work. The majority of these situations involved someone using very similar templates. And many (most?) of those folks had never heard of NVC! It sounds very artificial when reading it, but no one thought it was off. This is the exercise I recommend everyone does: Observe someone handling a stressful situation at work, and note how often they touch on all 3 aspects of NVC (specific observation, referring to the feelings, and the needs that are unmet).
Rosenberg didn't invent NVC. He just identified its existence.
No one's forcing you to use the template. Here's an example "So this is bothering you because it's affecting your efficiency?" This is perfectly valid NVC.
I think some of the problem arises in what is considered "good" communication. In the academic (and tech) world, we often describe good communication as concise and objective. This is great when communicating facts, science, etc. But in the workplace, we are not in the business of communicating facts and doing science. Most of us are in the business of making a product, that someone will buy, from which we will earn our living. And there are many competing motivations (some people want to "make" something, others are more interested in promotions, etc). Generally, we are heavily reliant on others. You can make a great thing but if the marketing folks refuse to market it, it will impact you (and vice versa). So "good communication" at the workplace involves dealing with a wider variety of people who have differing needs from you.
Becoming better at that type of communication is a lot of effort over a long period of time. It's easy to dismiss one type of style (e.g. NVC), but recognize that the "default" alternative is very poor and merely dismissing means accepting a poor level of communication. Had I simply read the NVC book and not dived deep into other books, I wouldn't have improved at all. It is because I decided that I will try to find a way to improve it that I recognized the merits of NVC and recognized its effective usage in the real world. It's also why I didn't stop at "crappy templates" and focused on making it less artificial.
Finally, NVC has a cult following. Ignore the cult. Just focus on the book.