Kate is only one of the considerations here. More important, I think, is Lili herself and her relationship with Fred. Neither the original book nor the revised version we saw in 1999 do much to flesh out any of the characters and their relationships. It's one of the reasons the original book is so poor. This was a huge missed opportunity in 1948 and in 1999, if they were revising anyway. The actual story of two divorced stars reuniting to do a Broadway musical of THE TAMING OF THE SHREW would be so much more interesting than the facile story that is presented in this musical.
It's similar relationship to that of Lily Garland and Oscar Jaffe in ON THE 20TH CENTURY, but think about how much more we learn about Oscar and Lily's history and their relationship with each other over the course of the show and how that relationship drives the entire show.
It's also why I have suggested here before that this story is ripe for a current team of writers to tackle and, rather than continually revising a show from 1948, investigate who these characters would be in 2019 and how this basic story would play out if, indeed, it were written today.