Yosselle Rosenblatts “Umipnei Chata’einu” begins the same as the traditional “Chayav Inish Levesumei”
The traditional tune for “Bnei Beischa kevatechila” in Chazaratz HaShatz is the same as the traditional “Keli Tzion…”. I’ve heard this observation attributed to R’ Chaim Brisk.