I don't see why an unfalsifiable statement is "meaningless". An unfalsifiable statement might be true or false.
It is true that an unfalsifiable statement may be true or false. But it is nevertheless meaningless. For example, the sentence "What will be will be" is a true statement, which is not falsifiable.
"Falsifiable" means "Logically capable of being proven false". When I say it is meaningless, I am not saying that it is gobbledy-gook. Rather I am using "meaningless" in the sense that the sentence doesn't provide any information about reality.
Here is another meaningless, unfalsifiable statement: "Everything in the Universe doubled in size last night." The contrary statement "The universe did NOT double in size last night is logically incapable of proof. Why? Because it would make no difference. A good test of this is to ask the question, "How would the universe be different if it hadn't doubled? In other words, if the truth value of a statement makes no difference to reality, it is meaningless. On cannot imagine a way to prove that the universe did not double in size last night, or any investigation which would prove that it didn't.
Now let's consider your statement: (1)"One can act without any regard for personal interest or desire." You say its not falsifiable. If that were the case, then the contrary statement, namely,(2) "No one can act without any regard for personal interest or desire" is
logically incapable of proof. But the statement IS logically capable of proof. One could imagine that every single person on earth acts only with regard to personal interest or desire." So how would reality be different if this were true? From your point of view and from mine, it would be quite different. Therefore your statment is falsifiable.
But this is not the case if we start with (2)""No one can act without any regard for personal interest or desire." I say that from the point of view of those who believe (2), the contrary (1)"One can act without any regard for personal interest or desire" is logically incapable of proof. For no matter what evidence you present, it is disallowed. You and I don't believe that (2) is unfalsifiable. You yourself provided a counter-example of a woman saving a child. But all counter-examples are explained away. Those who believe that (2) is true, cannot even IMAGINE a situation in which (1) could be proved. To them (2) is a tautology like "What will be will be." So (2), if true, doesn't provide any information about reality, if (1) cannot be proved in principle. Therefore (2) is meaningless (without meaning).