tabulature is really just used for starting out on guitar in my experience and it works because the alternative is SO approachable and SO instrument specific.

lead sheets are solving a whole different problem and almost ALWAYS have the melody in traditional notation in addition to the chords. some lead sheets assume you already know the melody and can transpose to whatever key is relevant.... but still a lead sheet is there to give you hooks in order to aide improvisation.

jazz musicians have not "split off", they have added some chord notation above the traditional stuff.