Because he usually doesn't need to.
I'm going to focus on the DCAU continuity, because it's the one universe I know the most about.
In Superman: The Animated Series and Justice League, when Darkseid fought against Superman, he could easily overpower him most of the time (episodes Father's Day, Apokolips... Now! Part II and even Legacy, Part II even if Superman managed to win in extremis in that last one). He truly lost only one fight in the Justice League episode Twilight, and it did cost him his life... until he's revived in the penultimate episode from the Justice League Unlimited.
In the very last episode of the Justice League Unlimited animated series, Destroyer, after his revival and his fusion with Brainiac, he became stronger, which allowed him to crush Superman with even more ease, until Superman decided not to hold back. But that didn't change much, because Darkseid still had something to put a stop to Superman.
Darkseid: It's called the Agony Matrix. Direct neural stimulation of pain receptors - all of them. Imagine the worst pain you've ever felt in your life, times a thousand. Now imagine that pain continuing. Forever. Oh, that's right... you don't have to imagine.
Still, that specific version of Darkseid fused with Brainiac had something special for the occasion (same video, 38 seconds forward):
Darkseid: Still alive. You impress me, Kryptonian. More... your valor has touched my heart. Oh yes, there is still some small part of me that knows mercy. I will end your pain... with something special I've been saving for just this occasion. [Darkseid draws a Kryptonite knife] I'm going to carve out your heart and put it on a pike in my throne room.
So, yeah, he's so strong that he doesn't actually need to use kryptonite, to the point that in the last occurrence, he pulls a kryptonite knife out only to kill Superman out of mercy.
As a side note, he's depicted as being so sure and so proud of himself that he doesn't even bother actually dealing with problems himself, usually sending his army to fight the nuisances in his stead. And even when someone dares to disrespect him, like Superman does, he usually uses his Omega Beam to avoid having to fight at all.
When you think about how he considers himself a god above everyone else, it would be pretty low from him to use kryptonite to deal with Superman.
Another point to add to the fact that it's not just about killing Superman to him: take a look at this excerpt from the episode Legacy, Part I, in which Superman was brainwashed into believing Darkseid is his adoptive father, and is sent to attack and conquer Earth.
He did it for the sole purpose of destroying Superman's reputation, so that the entirety of Earth would hate him, because he initially tried to persuade Superman into joining his rank to increase his army's power (which Superman refused); quote from the episode Apokolips... Now! Part II:
Darkseid: Think of it, Superman. The power you have now, it's nothing
compared with what I'm offering you.
Superman: You know I can't do that.
Darkseid: Pity. Still, if you won't be my knight, you will be my pawn.
Quotes from above are transcript retrieved from these websites.