That I recognize rights exist does not mean I believe they are necessarily self-evident to everyone. I recognize the rights, and I recognize that they are not self-evident to everyone. I do not know why you find that contradictory.
The three most fundamental rights, in my thinking, are the right to life, the right to liberty and the right to property. And yet even these can all come down to one right, the right to property because fundamentally the concept of rights is based on the notion of a human being as an individual owning himself. I realize this is a concept, an idea that one either accepts or rejects. But it is no less so than the idea that race does not make one person superior to another or any number of other ideas that seem self-evident in our society.
In Christian theological circles, the idea of self-ownership can be disputed with the claim that we all belong to God. And yet, there I would say even if one accepts that, God has still given us responsibility for own actions, and so we are at the very least stewards of ourselves, and so is everyone else, and to that end we are told to love our neighbor as ourselves and as one wants others to treat him so he should treat others. Thus, the rights still exist even if we are owned by God.
You mentioned the question of why we must recognize the existence of rights. Imo, to deny that such rights exist is to deny the nature of humans exists. If the rights, as I hold, exist because of the nature of humans, that we are individuals with individual bodies and minds and so therefore we own ourselves, then to say the rights do not exist means that we do not own ourselves. And if we do not own ourselves, then ownership of our lives can belong to someone else. Some people argue that humans are social animals and that we are therefore owned by society. They may not say it that way, but that is the essential meaning of the idea. I do not deny that humans are social beings, but I hold that society is not an entity unto itself, but merely the cooperation of individuals. And as such, society functions best when rights are recognized and respected.
I am explaining all this in basic terms of course, and as I said before, there are books written about this subject that explain, I am sure, much better than I can. I don't have time to write a book here, so basic terms will have to suffice.