Yes and no, I mean many of the parts were off the shelf and the vertical integration did slowly happen over time, but on the other hand iPhones never used off-the-shelf processor silicon. While they were Arm-designed CPU cores and PowerVR GPU cores at that stage (Apple would switch to designing their custom designed Arm-compatible cores with the A6 in 2012 and their own GPU design with the A11 Bionic in 2017), the SoC of the first few iPhones and iPod Touches were custom designed for them by Samsung.

Nokia did seem to rely more on off-the-shelf SoCs so having a custom design I think did contribute in some part to the iPhone's success. It's true that the software was a massive part but Apple were able to make different trade-offs with power consumption, CPU power and GPU power that enabled that software to work that might not have been as practical with off-the-shelf SoCs at the time.