I totally understand your position, and I completely agree that integration is way more complex, often doesn't work, and very quickly gets you in the Gamma function danger zone, but I still think it should be taught first (maybe just looking at elementary functions) because it's way more intuitive in my opinion.

There's a reason it was developed first, historically speaking.

Freshmen calculus I does not have to immediately jump into mechanically difficult integrals. You can leave those for Calculus II and concentrate on the fundamentals of integration and differentiation in Calc I.

You completely missed my point. I mean the actual mathematical analysis concepts involved in integration (measure theory, Rieman integration, principal value) are far more intricate than the concepts involved in differentiation.

Oh I see. That is indeed correct, but even a difficult book like Apostle largely concentrates on Riemann's theory for the introduction.

Learning should be like peeling onions. You peel off one entire layer, from all sides, before moving on to a deeper layer of complexity.