I'm not even entirely concerned with rights being apparently, at least at first blush, inconsistent.
What I am concerned with is what you've just done: to elevate a single right or a single set of rights to some priveleged status, simply on a say-so. These turn out to be both exceptionally arbitrary, on close examination, and frequently dysfunctional. Take a trolly problem, or a unit in battle, or a sinking ocean liner, or any number of other conditions, with your primacy of right-to-life, for example.
Note that life, liberty, and happiness are given in the US Declaration of Independence (though not the Constitution) as unalienable rights. That is, not rights which must be granted (any two-bit punk can claim your life, a zip-tie can restrict your liberty, and a wailing child can remove your happiness), but they cannot be given to another -- unlike your property, or business income, or even spouse and children.
So no, actually, I don't find that a convincing argument, though thanks.
Since you can never raise enough taxes to pay for everyone's needs, there's no consistent way to implement the modern notion of rights, so it simply becomes a list of irrational demands.
The only consistent system of rights (of rights which can be granted to everyone equally), take the right to life as a fundamental, and derive other rights from that. The derivative rights in such a system are the Enlightenment-era rights, I.e, life, liberty, property and the pursuit of happiness.