Karthus ult has to be long cooldown or he'd put unnecessary pressure on every lane and the jungle simultaneously, even if he isn't trying to do so on purpose. It's not like a TF ult or something where the threat of a tele is there until he picks someone whose not you to gank and then he goes in and your good to go for a few minutes, when karthus presses R he hits everyone else on top of his actual target as a side effect. it's like when janna and zilean were passively winning other lanes because of their global influence shifting otherwise evenly matched lanes in their favor without any effort on their part.
You can also kill lux to stop her damage, but if you kill karthus you may be baiting your team into getting aced since you wasted probably big cooldowns on him and hes going to continue doing damage for a while as his team continues to fight you.
Lux can ult to clear waves but she basically has to if she's trying to clear safely, and even then she also needs yo use her e to finish the wave unless ahead, and this costs tons of mana that she has to manage. Meanwhile karthus aoe farms at range with his q while regaining mana to the point where his q is basically his auto attack. You can't possibly deny karthus from farm or kills without having a dedicated assassin whose goal in life from level 1 is to dive karthus over and over, which is also super risky because of the wall of pain and karthus passive which makes going 1 for 1 even when ahead very possible.