It's not at all simple. And in your simple "explanation" you failed to answer the question. In reality, life began around 3.8 billion years ago and has continued in an unbroken line ever since. If the question is: Does a unique organism begin its existence at conception, the answer is a qualified yes,
Exactly what I would have said....the sperm and egg are every bit as alive as the zygote the moment after conception. Conception is the beginning of individuality and the beginning of a new path of development (as opposed to the different lifecycle of spermatozoa and ova), but life continues unbroken from one moment to the next. And what about identical twins? In that case, individuality begins at some point AFTER conception.
The ethical problem is the lack of an obvious moment when humanity (with the cultural and legal rights pertaining thereto) begins. Humanity is culturally conceived as a binary division between human and non-human. Life and development are smooth continuous processes without the kind of discontinuitues where rights could be unambiguously introduced. This falls on both sides of the "rights of the unborn" issue, where the only two obvious moments that could be appealed to are conception and birth. The fetus one second before birth is not any less alive or human than the baby one second after birth. And the single-cell diploid that results a second after conception is simply a merger of the two haploid cells that existed a second before conception; the life and material content of the cell is the same, although the merger creates an opportunity for a new trajectory of development. Unless we insist on one of these points as the magical moment when humanity and human rights springs into existence, it must lie somewhere in between. The problem is that because development is so continuous and unbroken, there is no obvious point in between that could serve as the dividing line required by the binary division; any proposed point would be altogether arbitrary. That leads to the eternal legal and social dispute, for neither the moments of birth and conception are truly discontinuous and there are no other developmental moments in between that could bear the weight of a binary division between legal and moral rights.
In short, our moral and legal systems like clear-cut divisions between categories but nature is rarely that neat.