It should be clear enough that "rights" are not a prime concept but a shorthand for the negative obligations of others. That is to say, "a person has a right to do X" really means "other people are wrong to prevent them from doing X". And this shorthand is useful enough that I use it myself quite a bit. But it runs the risk of being misleading.

When you say "people have a right to not be killed", you seem to imply that if innocents are killed, their rights have been violated. But that's inaccurate. My rights are not violated if I'm killed by a lion (who violates them?). Since the real principle is "people have a duty to not kill each other", and not "people have a duty to protect each other", there is no "right to life", except as a dangerously misleading shorthand.

Another place in which this language is shown to be harmful is in arguments about unemployent.

All people have the right to work.

If you dispute this, then it sounds like you're saying some people don't have the right to work, therefore you're saying it's moral to prevent some people from working (which is absurd of course even in the case of children). But if you accept the statement...

If you don't employ this person, you're depriving them of work and therefore depriving them of their rights!

Responding well to this talking point requires carefully explaining this point, which is not even feasible in some contexts (eg. on Twitter). And that's the problem with the language of rights.