The L2 and L3 healing limit break is very useful when it's needed but unfortunately it really can't be planned on so many people prefer not to use it even though it can save a run. I've personally used it to recover from Ifrit HM and post-heart Titan HM (although it's VERY difficult to time correctly on the latter), and even the level 2 has saved some dungeon runs like Stone Vigil with a not-so-hot healer and a Siren battle where a few party members didn't make it to the middle in time to avoid the room-wide charm.

I've also used the Tank limit break to keep the party alive during Titan HM's Stomps when he was at a sliver of health with both healers dead from bombs.

People just don't like saving it for emergencies I guess. Damage LB are extremely effective when planned around and are consistent so people just like to use it for those.