Yes, "no obligation to admit" means they don't have any obligation whatsoever, and that includes doing so for any reason they see fit and not having to disclose those reasons (if any) to you.
It is exactly the same as "I don't have to let you in".
For example, I do not have an obligation to let people into my house. I can choose to let them in or decline them entry. But there are certain preconditions I cannot apply. I cannot, for example, say "you may come into my house only if you murder my neighbour". That's because I'm legally bound not to induce people to commit murder. It would obviously be disingenuous to say this means I have an "obligation to admit" them.
It's the same with immigration. They actually are legally bound in certain ways - an immigration official can't assault you for instance. It's not hard to imagine them being legally bound not to search people's phones. That doesn't mean "they have to admit people".
You're confusing yourself with irrelevant analogies. You can say, "you may come into my house only if you give me your unlocked phone," and an immigration official can't assault you because there are certain protections granted to foreigners against being randomly assaulted. It's also not hard to imagine them NOT being legally bound not to search people's phones, and if you're trying to say someone's breaking the law here then it's your burden of proof.
Of course I can say that. I can say "you can't come into my house if you're black" too. The point is that it's unethical. It would be unethical for me to search your phone before you entered my house, too. This is not complicated, I'm not sure why you're having trouble understanding it.
The United States does have some rides about what border agents can and can not do. They can not sucker punch you, for example. They can request to see the contents of your phone and if you refuse they can choose to refuse you admittance into the country.
It’s not a question of fairness.
For what it’s worth I’m very much in favor of immigration and people visiting the United States, but this country and all others have the right to admit or not admit whomever they choose.
It is exactly the same as "I don't have to let you in".