I'm not opposed to
either the ideas of "rights", or of there being
conflicting rights, or of there being
rights of obligation, say, an obligation to help those in need (good sam laws, mariners' aid laws).
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.