The quest won't trigger. I have entered Novigrad, yet the gwent quest "High Stakes won't start. All other notices trigger certain quests but not this one.

Entering Novigrad does not trigger this quest. You have to either take the quest from a notice board ('High Stakes Gwent Tournament'), or find the scribe who's turning away a potential competitor in the tournament.

I have no mods. I have reinstalled and verified/repaired the game.
The things is: The quest doesn't even appear in my quest log when I read the notice about the Gwent Tournament on the notice board.

Go to the scribe.

Here's what I found out that (probably) triggers the quest: You have to enter the city, go to main square where they burn witches, witness the burning. After that cutscene, you can (probably) trigger the quest.