The problem is that it has been established with logic and rationality. It doesn't fit with your world view so you choose to ignore it and the thinking behind it. That doesn't change the reality of it - though you can stick your fingers in your ears, shout loudly and believe whatever you want to believe.
The logic is not logical. Because most humans think differently than other beings doesn't not make you, or me, special with "rights". For example:
I think that the reason human beings have rights is because we're self-aware... That, effectively, "earns" you your rights.
A lot of thinking going on... and some earning? Earning a birth given "natural right" is an oxymoron. An oxymoron is not logical or rational.
This is an argument that is based on what is "OK" being what is legal.
No it's not. In context, prior to civilization, there were no laws, thus nothing was illegal. Yet we were the same species.
"Legal" is something subjective. "Unjust" is not - it is something that is not just. "Immoral" is not - it is something that is not moral. Morality is based in logic - morality is objective.
'Just' and 'moral' are subjective. Unjust and immoral are therefore inherently subjective. Please, use a true axiom, not subjective logic, to make your point.
Society can deem things to be good or bad for society - subjective - and it can through government seek to have them made legal or illegal - subjective. It cannot deem something just or moral - they are objective and independent of society. Society exists to preserve and observe rights, not the other way about.
Since you don't
think morals are subjective, this isn't worth addressing. Your inability to understand people's morals are different, meaning subjective, is the underlying issue.
They don't come "out of nowhere", they come out of reason. Without reason, rights are not observed - that doesn't mean they don't exist.
Atoms exist without being observed. Air exist without being observed. Gravity exist without being observed. Your "reason" why rights are there (where is there?) without being observed has no substantial proof other than your opinion, and your opinion, or what you think (btw, this is the definition of subjective), has no bearing on reality.