You are absolutely correct - it was not a setup and it’s a disservice to Rosenthal and the marines as well, to say so.
And, I will add, Flags of Our Fathers is a fantastic book. I just read Flyboys by Bradley last month. Very good but not as good as FOOF.
I heard an interview of Rosenthal in the early 70s when I was a Marine photographer/journalist and the second flag was sent up later after the first raisers had moved back into combat. The colonel who owned the flag didn’t want it going to some general or admiral who hadn’t been on the beach under fire. A second squad was rounded up and sent up the hill to raise the bigger flag. Rosenthal said in the interview that they did more than one run at raising the second flag and that he almost missed the second set of shots because he wasn’t paying attention. Rosenthal provided an iconic vision of the event and he has received fame and recognition that will last a lifetime. The squad that gained the fame fought no less valiantly than the original flag raisers. The point is that it is too bad that the original flag raisers or at least those few who survived didn’t receive the level of recognition they and their families deserved.