I think you're exactly correct regarding this distinction, and I think that despite its ability to impel action, this has been a two-and-a-half-century-long disaster in the making.

You can't "find" a right. You don't trip over it, run some chemical analysis and look at under a scanning electron microscope and go, "Yep! We got a genuine 100% Human Right here!".

That means that the secular options for recognizing a right are limited. Religions don't have that problem, but they in contrast have no way to convince anyone else that their specific interpretation of what is a right is correct (and religious texts generally don't spell out what is a "right" very crisply so even among believers it's not always easy to decide). Furthermore, separation of church and state precludes the religious reasoning from grounding the granting of legal rights (and that has Biblical support from Mark 12:17).

Furthermore, you can declare whatever rights you want, and precisely nothing will happen. The universe isn't listening. If God is listening, the feedback is slow and subtle at best.

So you get vicious unresolvable fights over ideology and morality whenever there's anything less than complete agreement, and when the fight is over, you've accomplished nothing. You still need to decide what actual people will or won't do.

Thus, I think this was all a colossal mistake, except inasmuch as it was necessary to rally opinion around a clear idea (and maybe it was necessary). It simply doesn't reflect how we interact with each other and with the world. It's an abstraction, but more problematic than simplifying, unless the rights themselves are not the principles after all but derived from something else that you can actually reason about.

I explain what we should do instead here: https://ichoran.medium.com/responsibilities-are-more-fundamental-than-rights-69c80413d638