One other piece of this: it might help to distinguish between demanding that another person respect your rights, and respecting the rights of other people. It is hard (impossible?) to have a friendship in which one person is constantly demanding respect for their rights.
On the other hand, I think that as a friend, I also wouldn't habitually infringe on the rights of a friend. I won't generally sneak into his house and take things without asking. I won't threaten him with bodily harm, or harm him. I won't demean him or deny him agency or respect that is due. All of these things would fundamentally undermine friendship, and they are essentially violations of rights. Without an observance of these kinds of boundaries, authentic friendship isn't really possible. This is basically what I mean when I say that respect for rights is part of friendship.
So I wonder if part of the disagreement here comes from different focii? What I'm saying seems rather obvious and minimal to me. But the sense I'm getting is that when we talk about rights, other people aren't necessarily thinking about violations of rights, like the ones listed above. In the case where those kinds of violations are happening, I think it is perfectly legitimate to demand respect for rights, and to articulate them clearly even if one doesn't demand them...and clarifying these legitimate boundaries is a pre-requisite for genuine friendship. Friendship proceeds on the basis of a generally unstated, mutual respect for other peoples' rights. In cases where rights are being violated, though, I don't blame the failure of friendship on those articulating and defending their rights. I think the main problem is the violation itself. To lose sight of this is to engage in victim-blaming.
Having said that, I think that the Gospels and Paul articulate a peculiar and redemptive approach to rights, in which friendship is made possible even in cases where rights are being routinely violated. In those cases, what we find articulated is a voluntary laying-down of rights, which are also clearly articulated, in order to make a unique form of redemption possible. (Although, for this redemptive action to reach it's goal, there does have to be a voluntary acknowledgment of rights from those who violate them...imho.)
Still, this redemptive action doesn't make sense unless rights are presumed. Unless Jesus was, in fact, the rightful King who voluntarily opted not to use his power, then he was just a pretender, rather than a just ruler restraining himself from violence. If Paul was full of crap when he claimed a right to income from his ministry, then his decision to engage in tent-making wasn't an act of love...in fact, his claim to deserve something was obnoxious pretense, and his tent-making was mere necessity. Rights must be real, for the voluntary surrender of them to be meaningful (as opposed to pretentious b.s.)
Or to take another example of the consistent role of rights in the New Testament, we can look at Romans 13:6-8: "This is also why you pay taxes, for the authorities are God’s servants, who give their full time to governing. Give to everyone what you owe them: If you owe taxes, pay taxes; if revenue, then revenue; if respect, then respect; if honor, then honor. Let no debt remain outstanding, except the continuing debt to love one another, for whoever loves others has fulfilled the law."
There is definitely complexity to this, and we can't ignore that this was written by a man who repeatedly got himself thrown in jail. At the same time, the notion of giving honor where it is due, paying taxes when they are due (even to a corrupt regime), and paying debts are all, I think, quite sincere...which is particularly remarkable, given the nature of the regime being discussed. Here's what I think we find here (and modeled throughout the New Testament): It is not a critique of rights, or a warning against them, but an entering into a (deeply flawed) system of "rights" that fulfills them in a way that transforms them. What we don't find is stuff like this, "People who claim that you owe them debt are making grasping claims to rights that none possess, because all that matters is love, and not rights. There are no inalienable rights. Rights depend on duties, and what duties are these people performing? People don't need rights, or an acknowledgment of their rights. They need friendship instead."
Instead, we find a complex, sincere and incarnational engagement with existing systems of rights that transforms them through engagement, rather than negation. Rights aren't rejected. They are split open from the inside, like the shell around a seed.