In the end, have it decided not by us but by the product and the user.
First, set aside emotions and pride, and to judge once—using numbers— which product feels better for users, we’ll run a small experiment, create both options, and go for a quick pick. If even then I’m convinced there’s a mismatch at the worldview level, don’t force a settlement; clearly put on the table the options to split responsibilities or to part ways cleanly.