I've been talking with healer in raid and our dialog comes to the point that it's not comfortable to distribute healers raid saves and change it "on the fly" if necessary.
After that I've come to this idea - another way to represent timers during raid boss fight.
This is very rough concept that I've made, hope it's enough to show how it should work.

The top line is boss health with phases. Simple and straight.
Second line is phase timer. Because phases are repeated, it can' be represented as 1 lane with X counter of repeats (as example).
On this line RL and assist can place their own custom event for a certain player/players to do something.
I took as example Argus second phase: tank should split damage with someone and healer must cast raid save.

How it'd look on screen