It gets pointed out and poked fun at a lot that Kaworu should have just destroyed/thrown out the DSS choker, so both he and Shinji could survive. Here are three reasons that may have prevented this from being possible.
1. Saving the World - The choker was put on Shinji in the first place to prevent Eva Unit 13 from awakening and causing the Fourth Impact. The thought was that by killing one of the pilots, the Eva could be incapacitated. Kaworu knows this, and willingly chooses death over the destruction of humanity, as he did in the first series.
2. Metaphor over Practicality - The choker represents Kaworu’s willingness to shoulder Shinji’s burdens at his own expense. It also shows that responsibility can’t simply be discarded - Shinji’s choices and weaknesses have repercussions on others.
3. Death Wish - Kaworu has shown time and time again that death is always his fate and his desire. Whether it’s because his death is necessary to save humanity, or because he wants to quit living due to how he has been treated, he always gravitates towards this end result. So he was happy to use his death to help Shinji + the Lilim.