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So is it possible? I don't mean zero book keeping, but it would be nice if only 1-2 characters had to have stuff rolled for. I've gotten a bit tired or all the pregame rolling
This obviously implies lists with very little Psychic powers, which is tricky.

Top contenders include:
Belakor - no rolling needed cuz he knows set powers
Nurglings or Daemonettes as troops, although 1-2 Pink units wouldn't be horrible to roll for
Flesh Hounds - lots of them
Screamers - this is debatable as they need Psychic support
Soul Grinders

Are there any other units that are good "out of the box" without heavy Psychic support, or any Characters that don't HAVE to have rewards?

Plague Drones? You can build a decent deathstar with these puppies...

Basic Flying DP with only GR for the weapons? (nurgle prince with balesword, D-thirster)

Hellforged Hunting Pack? (which I'm building)

I'm following this one... the amount of bookkeeping does get a little tiresome

Kairos has a lot of random rolling shenanigans pre-game, but between his warlord trait and his reroll he prevents as much randomness as he causes I think. I'm also not sure that anything in the book is tough enough to run without psychic support - Flesh-hounds and Screamers might work for you but they're not spammable, so they need Cursed Earth etc

If you want something a bit more creative than Screamerstar/ Flesh Hounds/ Nurglings then maybe KDK is an option?