That's not true; under the constitution of my country, which is a republic, rights are based on the sovereignty of the people.
A more relevant issue is: does it matter? My country didn't transition from a monarchy to a republic because the king was convinced by strong reasoning to abdicate; there was a coup d'état, and he was forced into exile.
However as you point out - using 'sanctity' type special status cut's both ways - what pretty much every other country sees as sensible action - control of lethal weapons - is blocked in the US, partly by, by pointless debates on 'rights'.
> If you surrender the concept of natural rights, you lose all of your negotiating power, as citizens.
True power comes from collective action - the Baron's ganged up on the king to force the Magna Carta - it wasn't the power of the philosophical argument - that's a cover to allow face saving ( though, of course, face saving is essential to non-violent progress ).
It's important to note that the recognition of natural rights was critical in the argument against monarchy, if this concept of natural rights is eroded or replaced in government, the concept of negotiating rights and/or civil disobedience becomes very different in philosophy, because it would mean that rights are granted by whoever's in charge of the government, rather than being an emergent property of humanity that the people agree to limit for the greater good. If you surrender the concept of natural rights, you lose all of your negotiating power, as citizens.