Two points to taking out the shipyards:
1) It will also destroy any ships currently under construction or in refit.
2) The shipyards can be rebuilt, but will start at minimum size (1,000 tons) and it will be years before they can build large ships.
Regarding the Armageddon option:
Certainly worth doing with the remaining missiles once the shipyards are destroyed, but it will probably be rather less effective than you assume.
Against a Prix or Rakhsha planet (was it in Lalande 21185?), a full 2,000-3,000 points worth of missile damage created a nuclear winter that raised the colony cost by about 4 or 5 points. If the planet didn't have spare infrastructure (which they might indeed have on their home world, to send out to colonies) that will cause a gradual die-off in the civilian population. But presumably they'll switch their factory production to infrastructure before very many pops die?