On my first playthrough I didn't experience any major problems, except for one locked chest apparently without a key, and the occasional crash. But I'm playing NG+ now, and experiencing a number of problems.

Contract: The Merry Widow doesn't update when I take the skulls, so I'm unable to complete it.

Blood Ties doesn't update when I pick up the letter to the guy's mother, so I'm unable to complete it.

Legendary Griffin Steel Sword and Silver Sword diagrams are both missing, so I'm unable to complete that quest (and also unable to get the diagrams).

A lot of chests that had good stuff in them in the first playthrough are now unlootable/empty. Sometimes saving and reloading that save works, but it often doesn't.

I've tried saving and reloading for all of these issues, but it's only helped occasionally for the last one. I've also tried verifying/repairing my installation. Is there a console command I can use to advance quests past these problems?

I started NG+ at level 89, currently level 105. The mods I'm using (was using all before starting NG+, but not most of them before 1.21 update):
90k Encumberance
AlwaysFullExp
BetterLoot
CrossbowDamageBoost
LevelingDLCArmor
FullMonsterExperience
GwentCardDealer
ImprovedAxii
LevelingGrandmasterGear
MultipleMutations
SlotsSlotsSLOTS
LevelingStartingArmor
LevelUncapper
Wealthy Vendors with Fair Sale prices

Guess the next step is to try removing some.

Yep. That should have been your first step. With that amount of mods, many using scripts no doubt, it's no wonder you're having issues in crossover (NG+) games.

Wasn't my first step because it's a complex one, and there might be simpler answers.

And? What's your point? You want the game to work, don't you? There's none simpler than removing and checking, especially if you use a mod manager.

Aye. With my mods, if possible. Not using a mod manager. They seem to cause problems for a lot of people. Just script merger to resolve conflicts. Even if I were though, removing a mod, re-merging the scripts, testing to see if the problem is resolved, then repeating for each mod, and then combinations of mods, is a lot more complicated than a console command, or even a "known issue; cdpr are working on it", or a "known issue with x mod".

I tried removing several mods (leveling starter gear, dlc gear, grandmaster gear, gwent card dealer, improved axii), no luck so far. But I was able to deal with the Merry Widow and Griffin gear quest issues with console commands. Could probably do the same with the other quest if i could figure out what item to add. As for the chest issues, well, I'd rather have the mods if I have to choose.