I don’t believe they excluded other forms of genesis as well. This is how I recall it.
Then why can you kill them on Shabbos? The point was that they don’t reproduce normally, so killing them is permitted.
If some reproduce normally and some don’t, then I would think you have a safek (did this particular louse come from an egg?) and you would have to apply the rule of safek d’oraissa l’chumra.
I also am not so sure “lice” is the correct translation of the organism that is being referred to.
I have never heard of “kinnim” being referred to as anything other than a louse.
The Wolf