IMPORTANT NOTE: Like stated in the previous paragraph, all Covenants are very close to each other in terms of power. Outside of absolute min-maxing, feel free to pick the Covenant you enjoy the most.
Necrolord is an incredibly strong pick for Mistweavers in raids. Bonedust Brew is a great ability which provides a fair amount of extra free healing. Being on a 1-minute cooldown by default (and there is a possibility to reduce the cooldown even further with conduits), it fits most of the damage patterns in Castle Nathria really well, especially on later bosses. On top of that, picking Necrolord gives us the access to Emeni - an incredibly strong soulbind due to Lead by Example trait, which gives a large amount of main stat to us and a smaller amount to our nearby raid members whenever we cast Bonedust Brew.
Kyrian is also an extremely solid choice for raiding for Mistweavers. Kyrian will be your choice if you really like Fistweaving as it makes Invoke Chi-Ji, the Red Crane much more potent, but even outside of that synergy Kyrian is extremely close to Necrolord in terms of power, while being a more versatile choice, which works with all talent builds. As for soulbinds, Pelagos looks like the better option compared to others at the very early stages of the expansion, while Kleia becomes a better pick later on.
Mythic+: Kyrian / Necrolord
In Mythic+, both Necrolord and Kyrian are slightly ahead of Night Fae and Venthyr (however, as stated before, they are not very much behind and still remain very reasonable picks if you wish to do so), and Kyrian has a slight edge over Necrolord. The reasoning for this would be the fact that Chi-Ji is a very solid pick for dungeons and as mentioned earlier, Weapons of Order empowers our Crane friend by a fair amount, and also because how Bonedust Brew loses a bit of its value in a dungeon setting. However, Necrolord can still be quite useful and provide some extra damage and healing, which is always great to have.
Soulbinds and Conduits
Potency: Jade Bond
Slightly increases the healing done by our Celestials and reduces their cooldown every time we trigger our Mastery healing. Overall a very good choice if you can utilize the Cooldown Reduction aspect and get an extra use on the fight or get better timings for your Celestial.
Potency (if Necrolord): Bone Marrow Hops
Potency: Resplendent Mist
Provides a nice boost to our Mastery healing and makes chasing Essence Font HoTs more valuable. Not a bad choice at all, but two options mentioned above provide a bit more value in my opinion.
Potency: Rising Sun Revival
Will provide around 10-15 seconds of cooldown reduction on Revival if you cast RSK on cooldown. Generally is a good choice for an extra chunk of healing from Revival since in good circumstances it can provide around 2% of total healing done.
Potency (if Kyrian): Strike with Clarity
Was carrying Chi-Ji build previously due to enormous amount of extra Mastery you’d get, after the nerf it's quite underwhelming: Weapons of Order duration by default is already longer compared to Chi-Ji (30 seconds to 25), so the increased duration part doesn’t matter too much, and the amount of Mastery you get now is extremely low, which makes the conduit very weak. However, it’s worth keeping an eye on it in case it will get buffed in the future.
Endurance: Fortifying Ingredients
The best Endurance conduit we currently have the access to; it provides a nice absorb after casting Fortifying Brew.
Endurance: Grounding Breath
Underwhelming compared to Fortifying Ingredients, but better to use than other options in case you have 2 endurance slots for some reason.
Finesse: Dizzying Tumble
Not by any means groundbreaking, but gives a nice damage reduction from enemies affected by our Leg Sweep for a short period of time. Important to note, that the effect is activated after the stun duration has expired.
Finesse: Swift Transference
Whenever you only have enemies that are immune to stuns, use this. The effect is almost negligible, but it’s better than other options (read as nothing).
The best legendary for raiding is currently Ancient Teachings of the Monastery. Pretty straightforward: After Essence Font, for the next 15 seconds your single-target damage abilities will heal an injured ally for 250% of damage done.
Assuming that Monks should be spending quite a fair amount of time DPSing with having Rising Mist talented already, this legendary provides a big healing done increase for almost no additional mana spent. An important thing to note - the buff is gained after Essence Font channel ENDS, not the start, but cancelling the channel or getting CC'd during the channel will give you the buff regardless.
Invoker's Delight - Gives a big haste bonus for 20 seconds after summoning your Celestial. It can provide some big value since all healing from Enveloping Mist and Enveloping Breath scales off haste, and ultimately having a second Bloodlust gives you a window of enormous burst healing. Definitely a solid option, and the go-to pick on some bosses (Sun King Salvation, for instance.)
Tear of Morning - Empowers our Vivify cleave healing by quite a significant margin, as well as making Enveloping Mist healing cleave in the same manner. Provided that we are already encouraged to cast Enveloping Mist in the raid a lot more because of the new Celestials and Enveloping Breath, this is a very nice bonus.