I never use Saber dance (don't see any need for it frankly) but it does cancel Fan Dance effect. I'd much rather have Saber dance and Fan dance share a timer to get a timer back for something useful, such as splitting waltzes.
Other jobs would have similar benefit from moving shared timers around. For Thf, sharing a timer between Mug and Steal comes to mind (as long as the duration doesn't go up to match that of Mug - come on, 15 minute timer on a skill that is NEVER used anymore?) in order to split Steal and Aura Steal into two separate actions (with a reduced recast on Aura steal).