Quite to the contrary:
On closer examination your position demands it. A newly combined egg and sperm are being lobbied as falling under the umbrella of "right-to-life".
It's the point of beginning for a new entity, one containing as it does everything necessary to continue down that chronological pathway to the inarguable fruition of you or me. Nothing with that potential exists before the point. The question then becomes at what point along the chronological line of being will/should/must right exist?
By comparison you employed your son (among others) as obvious demonstration to such right and asserting that the sperm/egg co-mixture assumes (as per human life in general) the same right your son enjoys.
Rather, what I've stated is that the moment of conception is the beginning point where the vestment of right is as arguable as not. Every point up until birth is similarly situated. The argument follows the implications of that in relation to the foundation of the compact involving right.
So I don't attempt to answer on the point of whether the act of conception establishes right or whether it's breath, or brainwaves, or heartbeat, or the ability to exist independent of the mother, etc. None of those positions self-evidently establish right. My catch-all is necessitated by the potential for the right in each moment beginning with conception and the understanding that we have no right to abrogate that right wherever it exists absent fairly grotesque violations of the compact that are impossible for the unborn to have accomplished, excepting the mother also has a right to her life and to defend it against a direct threat posed by the unborn's presence under certain and rare circumstance.
The implications are obvious...you're claiming no difference between your son and the sperm/egg i.e. A=A; they both identify under the rubric of "life".
No difference in the one aspect that controls what we can or can't do to either.