Well that is probably correct, if the embryo is living it is human and the human is generated - in a sense. However, in generation only the physical aspects of man are passed on from one generation to another, there is no genetic " soul " material to be passed on. God implants the soul, man does not " pass on " a soul. So in para 404 of the CCC it says “…Still , the transmission of original sin is a mystery that we cannot fully understand…”
The body of man has no effect on the soul except to provide the soul with the " tools " necessary for it to be and to act as the substantial form of the human being. It is the soul which is the source of all activity. But the soul was created by God, it was not generated in the sense it could be passed on to another generation. So, how was sin generated?
My big idea. God forsaw the sin(s) of all men ( except Mary and Jesus ) and therefore created them without sanctifying grace ( privation of the special grace Adam and Eve had originally). So in that sense Adam and Eve would have generated men who inherited original sin ( in the sense that their soul was created is a state of privation in regard to scantifying grace). Although, properly speaking, only the body is generated as the soul is implanted - unless you want to give a special definition to " generation. "
Or, as para 404 says, Adam represented the entire human race and what he did would be the fate of all. Since he sinned, his progeny to the end of time would be generated without sanctifying grace ( Mary and Jesus excepted). But still it is only the body that is generated unless you want to give a special definition to " generation. " The soul is implanted, it is not generated. But that is man’s definition, perhaps not God’s and maybe not the Chruch’s either.