People have already mentioned shamans being miserable in exp so this post is going to focus more on their 6-man and raid dilemma.
Giving my 2 cents as someone who never mained a shaman but has raided on one a bit and gotten hard carried by various shamans on hard content for a decade straight. The class is in a weird spot because its strength is situational and dependent on the content being done.
Shamans are one of the strongest classes in the game when it comes to doing difficult, on tier, tightly tuned raid and 6-man progression content. Part of their niche defining strength is the ability to chain woundbanes on a tank without OOMing for far longer than druid or cleric mana can sustain. This means that shamans can allow clerics and druids to spend a much larger portion of their mana on group healing throughout a fight. Shamans provide another set of group heal procs and clickies to their group (this is an often overlooked strength, it allows the cleric or druid to save mana on group healing). The other large part of shamans niche defining strength is their damage. Scourge of Life and Caress of Sivyana both have good damage per tic AND damage per cast time, meaning good shamans are capable of getting high value from their DoTs if they have spare time between heals. This value is even higher if the shaman blood rituals the DoT. Shamans also have the option of ignoring healing entirely and fully committing to DPS. By doing this, they have the potential to do damage that approaches what a real DPS class can do. Having the flexibility to do high damage when needed as a healer is incredibly useful for some of the games hardest encounters. A lot of encounters might feature a burn phase or a high priority add that must die ASAP. At low gear levels (and low levels of familiarity with a fight) you struggle to meet damage checks and your shaman fully committing to damage while letting other healers carry the healing temporarily can be a huge benefit to the raid. This is particularly good when you consider that a lot of these damage checks can overlap with when you'd want to use Call of the Blessed.
All that being said, shamans strength fall to the wayside and become significantly less relevant as your raid gets more gear, exp, and familiarity with a fight. Nerfs to an encounter or changes that make the game easier in general also contribute to this. As your raid gears up the tanks will take less damage, fight times will become shorter, and other healers will get more mana. As clerics become more familiar with a fight as they re-kill it in farm and learn the damage patterns they also become more mana efficient. This is by virtue of timing group heals better, overhealing less, and learning when you have windows to proc your group to full instead of ghealing. It doesn't matter if a shaman can heal for 20 minutes straight if the fight is over in 6 minutes and none of the druids or clerics are in danger of running OOM. Shaman damage also starts to matter less when content becomes farm as damage checks that used to have razor thin margins get completely blown out of the water (and your raid becomes capable of living through the scary phases longer / more easily). If not for the recent buff changes you'd likely drop shaman from your raid in a optimal composition when doing most farm content and replace them with another DPS or a druid (depending on if your raid needed another healer or another dps). In easy content / farm content you simply just have worse healing throughput than any other healer but possess the ability to cosplay as a DPS once you've recognized you could cast literally 0 healing spells without it causing the other healers in the raid to OOM.
The issues shamans face in farm raid / 6-man are for similar reasons to their low desirability in exp. Their healing throughput is low relative to other healers and requires a lot of globals spent healing, especially because: They lack a group heal outside of clickies and procs. Shamans struggle in solo-heal situations because of this. They also can't blood ritual to boost healing throughput or damage as safely as they can with a 2nd healer present. DoTs lose value in exp where mobs have low health pools. They don't have access to strong AE damage.
Strong mana sustain through Canni5 doesn't matter in exp where any class can either med or get fed infinite mana by a necro. A potential suggestion would be giving shamans a low efficiency but high throughput healing spell, allowing them to do more healing in situations where mana is less of a concern. The class could also use an upgrade to canni4 now that CMal bp is nerfed. A cannibalize spell with high mana return but a short cooldown would be worth casting without forcing shamans into feeling like they need to commit every spare global to casting canni.
The issues of DoTs being bad in exp and not having strong AE damage are moreso issues with the game at large. Over the years it seems like AE damage has been nerfed time and time again but it still remains the most efficient way to grind by a significant margin because no competitive options to do "hard" or "single target" exp exist. The obvious bandaid solution to any of the classes that are currently bad in exp is to just give them a competitive way to do AE damage.