For various reasons I am reading again the DitV game manual these days, and reading again the chapter about the initiation conflict made me think about the utility of that conflict.

I think that the principal reason for playing that first conflict is to show the players how to play the game, and how conflict and fallout works, and to get a first taste of the setting, before they are dropped in their first city. When I played most of my DitV games, I always played these conflict at the start of a new group of dogs, but thinking about it now, I can't see the utility when I play with experienced players. It force people to wait until everybody has one, and it don't teach them anything they don't already know.

So I think that in the future I will skip them, or use them only with new players who still had not played DitV.

My question is: am I missing something? There is something more about these initiation conflicts that make them worth the hassle even with experienced players?

Yeah, the Initiation conflict is not a procedural necessity, just a supportive tool for learning the rules and getting a sense for the setting and characters. Experienced players still benefit by learning something unique about their characters, but if you don't need this extra insight into the character, then the initiation can be skipped.