The first is to reduce the view that the healer's main job is to fix other players' mistakes. The way to fix this is simple: include debuffs as part of most avoidable damage. The easier a mechanic is to avoid, the more severe the debuff, though this doesn't have to be a rule.

Petrify, stun, paralyze, and pacify/mute. Most mechanics apply one of these status, depending on the difficulty it is to avoid. Extremely difficult things only give a pacify or mute. Perhaps both. The easiest mechanics to avoid give a petrify. The debuffs don't have to last long. Three seconds for particularly debilitating debuffs, five to ten seconds for the lesser ones. They can even be cleansable to give particularly good healers additional ways to contribute, but they punish the one who messed up first and foremost, and in a way that everyone can see.

The problem with normal avoidable mechanics is that the one who's punished by them is always the healer. The one who made the mistake is only punished if they die from it, and that only happens if they get one-shotted, or receive so many vulnerability debuffs that either the healer can't keep up, or they get one-shot.

There's not really much leeway for these sorts of mechanics, since most of the time people don't care much about taking a hit unless if it'll immediately kill them. Meanwhile, the healer is constantly punished for others mistakes, as every time they get hurt, the healer has to take time and resources fixing them up. Time and resources which could be needed to clear the raid. Both between enrage timers and mandatory healing mechanics.

But...the only heal mechanics that are mandatory are raid-wide AOE?

Personally, these are the most boring forms of damage to heal. Not only have tank-busters fallen significantly in frequency, but there's also not that many mandatory hits that don't just target everyone. Pin-point damage is much more interesting to heal, though knowing that all your single target healing is done because someone else messed up is far less gratifying than knowing that it's a boss mechanic designed just for you and your co-healer.

So, aside from the first change, I believe that if SE is serious about making healers more about healing than being supplementary DPS that covers for others' mistakes and occasionally spams heals to cover raid-wide damage, the other change that should be implemented into the duties is a large amount of targeted damage. Not just targeted AOE markers that someone needs to bring away from the group, but for mechanics to do meaningful damage.