In 5e most undead have a decent list of immunities or resistances. Taking the revenant as a conceptually sort of PC friendly model they get:
Damage Resistances necrotic, psychic
Damage Immunities poison
Condition Immunities charmed, exhaustion, frightened, paralyzed, poisoned, stunned
Yeah I always personally felt like the 5E list was excessive and made some limited sense for powerful non-free-willed undead, but none at all for free-willed ones, particularly not ones with semi-operative corporeal bodies, and where those conditions could be applied from sources that, conceptually, would mess with a ghost or wight or whatever.
So like taking your handy listing, I'd say, even NPC undead should almost certainly lose:
Resistance to Psychic damage
Immunity to Charmed, Frightened, Paralyzed, Stunned
None of that makes a huge amount of sense, and some of it is actively perverse - I mean, in fiction, free-willed undead certainly have emotions, and typically can even feel fear and like/dislike things - especially bloody Revenants! Paralyzed and Stunned should absolutely be possible to inflict on most corporeal Undead, esp. as they're often from highly physical sources.
There's no earthly reason Psychic damage shouldn't work on Undead of any kind. If anything it should be one of the more effective kinds of damage.
Re: poison I note that the PC construct/undead template already has that:
- You have advantage on saving throws against being poisoned, and you have resistance to poison damage.
Personally I think for corporeal undead with semi-functional bodies like Revenants and Vampires they should have resistance not immunity (esp. as "vampire drinks poisoned blood" or similar is a classic fantasy/horror trope). For your Lord Soths and Zombies and so on full-on immunity to both makes sense. Exhaustion immunity makes sense for NPC undead (esp. Revenants who are legendarily relentless), but given that PC undead already have reduced sleep is probably unnecessary (I doubt it would break much RAW, but I know a lot of people have house rules making more use of Exhaustion than RAW does).
So the only remaining question is Necrotic damage. It's a fairly good Resistance to have, but not nearly as good as Fire damage (23 monsters in the MM do Necrotic, vs 46 in the MM do Fire, based on what I can find). Could take or leave.
I think the D&D of ancient times (pre-OD&D!) and 4E both hit on the right approach for someone wanting to be a "serious" undead, which is to make it a class (or in 5E, probably a subclass*). Ancient D&D had Vampire as a class (indeed the Cleric class was created in reaction to this), even if it never made it to a rulebook (but OD&D specifically says "Yo if your player wants to be something wack and you're into it, go for it!" - giving a dragon as an example, just saying they should have to start weak and get strong, like other classes). 4E's Vampire class was solid if not particularly optimal.
* = I dunno if it was a UA but I feel like there was one already.