Is initiative dumb? No. It's just a type of mechanic that can be used poorly or well or be unaddressed. Asking if initiative is dumb strikes me like asking if 12 sided dice are dumb. There is no one standard best practice, perfect set of mechanics that makes the best game. There are a variety of games that are entertaining in different ways, for different reasons, including the way in which their mechanics interact and how they shape the play experience.
Why not let the PCs go first all the time?
What are the goals of the game design, and does letting PCs go first all the time serve those goals, run counter to them, or just not matter? I fully would expect this to vary based on the game. Like, an Old West game where players got to go first all of the time would take some of the edge off a gunfight, where the Westerns themselves make a big deal out of the contest and the protagonist doesn't always get the shot off first. On the other hand, if you were playing Vampire or some Demi-God style game (Godbound, Exalted), and decided PCs always went before mortals... that wouldn't seem crazy. It would reinforce their inherent superiority. So, my answer to "Why not let PCs go first all of the time?" is that there are times it would undermine the vibe. Not in every game, but certainly in some.
How does that hurt the game aspect?
Again, this varies greatly, depending on the game. I'm Mr. "Street Fighter", and I can tell you that game is about beating people up like in the video game, but with crazy adventures along the way. The initiative rules are core to how the combat functions, tactics, guessing what your opponent is up to and that makes it enjoyable. If you remove it, you destroy the "Game", and are just left with a mediocre incarnation of the Storyteller system. It would be like removing Sheep from Settlers of Catan. On the other hand, where games aren't about that, other initiative approaches make more sense, even as loose as Apocalypse World-like, "just decide what makes sense and kinda work it out as you go".
I think rolling for initiative is archaic. It worked once but now it's dead.
Initiative regards the determination of who gets to act, when, and how often. Even, "Players always go first" is an initiative system. Rolling for initiative is only one of the ways initiative can be determined. For the most part I'm not a fan of individual rolled initiative, but it's not the worst thing in the world. (That would be individual initiative, rolled every round, bleh). Overall, though, my point of view is that there is no inherently "dumb" or "archaic", divorced from the context of the rest of the game. An initiative system can just be someone's cargo cult-like afterthought tossed onto their system, without them thinking about why they're adding this particular amount of overhead and tracking. In that case, it could very well be dumb and archaic. But... it's not something I can judge in isolation from seeing how the rest of the game works with it.