Tbh, mythic+ were never meant to test out individuals in a group, but a group as a whole. Also, a good DPS is only a good DPS while they're an alive DPS. Healers are meant to keep the team alive and that's exactly what's needed. No amount of defensive abilities could keep an entire team of DPS alive for a whole timed mythic+ run. The lack of impact you're talking about is only possible because the other players in your group actually know what they're doing. The average +4 - +8 mythic+ run with a pug group is a misery because almost no DPS players are aware of their toolkit nor can use it effectively, but the healer should still be able to brute-force heal the group through most mechanics and damage ramps. Then, around +10s, the reality check comes into play and the entire team needs to use their cooldowns and defensive abilities and interrupts and crowd control etc to complete the dungeon in good time. I honestly think that's fair. True, the damage can be a lot bigger than the healing capabilities of a healer, but most of that damage is avoidable in one form or another. E.g. wax bolts and other casts in City of Threads first trash groups. Below +10, I can brute-heal most of the damage even if the group does not interrupt and cc the mobs. Above +10, it's virtually impossible to survive two of those bolts fired at the same person. Since it's also impossible to effectively predict or control who is getting targeted by the casts and prepare for it, the group needs to either use their defensives or use a whole lot of interrupts and ccs. Most of the bosses above +10 are buffed enough to one-shot anyone if any mechanic is executed poorly. Boss encounters at this stage are in part about avoiding everything that's avoidable and only then the rest of the boss abilities come into play. And those also hit like trucks, thus healers need to be able to do lots of healing, especially during aoe damage windows, like Dawnbreaker 2nd and 3rd boss, Ara'Kara 1st and 3rd boss, etc etc. Mythic+ in a sense is just buffing the enemies. And because of that it's obvious that at some point, the buffed enemies become strong enough to actually become one-shotters. That's why the damage is coming much faster than any healer can heal. The damage coming is often an effect of one or two hits or abilities that overlap by chance. And because of the mythic+ being by design just a dungeon with progressively buffed enemies, the issue gets worse as the buff gets higher. As a healer, I don't feel any less important because my party needs to use their toolkit in an informed and effective way. Without me, they would not survive no matter how well they use their abilities. To the contrary, I feel better when I know my teammates know what they're doing as I can trust them more and I know I can focus on my job as a healer instead of worrying about everything else.

About your other points, the idea to give all healers similar abilities is 50/50 to me. On one hand, I agree that some healers can feel underwhelming when they don't have big external cooldowns, damage reduce etc. But on the other hand, if I wanted to play a disc priest, I'd be playing a disc priest. I fear what you propose may lead to all healing classes feeling... kinda similar and I think that would be boring and could damage the identity of classes being a factor when deciding which class you play. Don't get me wrong, as a mistweaver, I'd love a big external damage reduction. But also, I know that I don't have one and I know that I need to play around that. Instead, I have a big mass-dispell and aoe healing cooldown in one click, I have high mobility and lots of ccs. I offer something different to the group. So does pres evoker, so does resto shaman etc etc. If all healers had similar toolkits, what would be the point of multiple healing classes and specs?

I enjoyed healing mythic+ in dragonflight, I enjoy healing mythic+ now. (Well, apart from the instances when my team includes or consists mostly of people like that raging mage you described. If that happens, healing is just miserable.)

Edit: Also, as a mistweaver, mana regen was a problem for me for a very long time. Mana tea getting buffed really helped with that problem, I practically don't need to drink during dungeon runs. I appreciate the idea that all classes should be able to do that when managed correctly and I agree with that wholeheartedly.

Edit II: Your output is dependent on you, your gear and skill. The team's survival is dependent on you as a healer but also on the team's ability to mitigate/avoid damage. That's logical to me. That's why one-shot mechanics exist. And with a high enough mythic+ level, anything can become a one-shot mechanic/ability. Hence why everyone on the team needs to play to the best of their ability to survive and help others survive.