I dont believe its your healers under-performing (aside from their deaths). There are things you can change about your execution and CD usage to improve your success. They probably can always do better regardless, but making the encounter easier overall will greatly improve their own performance as they will have an easier time focusing.
you still had warmages up in phase 4 on some of those tries contributing a lot of damage (these should die in the intermission, you can have the reaver up still on phase 4 and tank switch, it will save the raid from taking a lot of spike damage).
Your dps could focus adds a lot better to reduce the time these adds are hitting tanks.
As you are only going 4 healers I suggest aoeing the small intermission adds once warmages are dead (and all of them have spawned, the volatile anomoly ones) with CDs up (SLT+HTT/Barrier/rallying cry) otherwise go with 1 extra healer as those adds dying continuously during the intermission can be a drain on mana and it is a long fight.
on the 11:05 try your shaman used SLT once and HTT once which is wasting at least another chance to use it effectively. You would definitely benefit from assigning all of your raid CDs prior to engaging the boss.
Looking over your players death logs might also help you as loosing healers at an time is going to put more pressure on the ones remaining alive.
For phase 4 alone, not having warmages up, reducing personal fails causing deaths, have 2 healers (priest and shaman/druid) focus the tanks while the other 2 raid heal and make sure your tanks and raid are getting the CDs they need will greatly help just in that phase alone.