You're describing two systems with the same end result - rights, but with exceptions and limits. One of the systems is simply more honest about having such a setup, and it's not the American one.
"Rights, with no wiggle room for edge cases or accounting for conflicts between rights" is "wrong by design". So's a giant, ever-expanding defined list of edge cases; I've run a popular web forum and that one leads to "ah but I didn't do exactly the thing listed in Article 73 section 48 part ZY" nonsense.
Our legal systems permit the resolution of conflicting rights and responsibilities. I'm glad that exists, but I'm also glad the Europeans took the American Constitution's failings and made it clearer that it should exist.
"Rights with wiggle room" are not rights. When someone else gets to decide that your "right" will be curtailed because of some "edge case", that's known as a privilege, not a right. Likewise, your concept of "rights" is flawed if it includes any possibility of conflicts. Real rights—negative rights—do not conflict with each other. If an action would infringe on someone else's natural rights then it's out-of-bounds without their consent. If not, it's permitted. Simple and entirely conflict-free.
There are flaws in the American Constitution but this is not one of them. The failing lies in the courts, for making up exceptions contrary to the law and to natural rights, and in the EU for codifying those flaws into its own legal system.
I have the right to free speech, but your rights mean I can't exercise it by shouting at 3am in your living room.
I have the right to free exercise of religion, but I'm not allowed to exercise the Sacrament of Nuking Chicago as part of my services.
Deciding which rights win out in these scenarios are where the courts come in. Europe acknowledges that; the US kinda had to "discover" it as an option and invent space in the Constitution for it in spots.
Being wrong by design is worse than having some implementation issues.