Zenitsu wanted to become Douma's tsuguko and learn ice breathing as a way to show his his gratitude and thank him for everything he has done for him but Kaigaku wouldn't let him because of how dangerous it is
Remainder; Ice breathing is dangerous even for the users themselves as the nichirin used for this breathing style could kill them if they don't have the necessary qualities to master it correctly, which are; speed, flexibility, advanced hearing and reflect and they have to be able to make calculations as quick as possible.
While Zenitsu does count with an incredible speed and hearing, he lacked the other qualities needed to master it, as this others qualities where not his strength