When performing popular songs, mellow music, love songs and old favorites, is it necessary for any reason to play extend chords beyond 9th chords to sound good?
Most pop type songs will include major and minor triads, sus chords, 6th chords, the 3 main 7th chords, and occasionally 9ths, more often dominants than minor 9, #9 or b9ths So, as leftaroundabout states explicitly, no. 11ths and 13ths can be used, in place of others, to spice up the accompaniment, but that's discretion, rather than what's written.
'To sound good' is subjective, but with the options of different voicings, slash bass notes, etc. there's probably enough to be going on with, without straying into jazz type territory, which almost expects chords to have bigger numbers than 7 or 9!!