letting go of his name and his ties to oden
is not physical death
but its "death" in another way.
the point of killing momo , luffy , oden as leaders of the alliance was to crush the warrior spirits. if kaido could get luffy to come out and say he gives up , it achieves the same purpose for instance.
momo abandoning his name likewise achieves the same purpose because ultimately he is the one they want as shogun. This is also a question that you cant answer till kaido's flashback is fleshed out , in his drinking spree kaido has revealed his failure to protect anything amongst other deep regrets he has. To reduce his suicidal tendencies and alcoholism to the result of the oden bout would be false.