The bullet nose, in comparison to other factors, has little to do with accuracy. If you've ever shot "long" plastic tipped bullets, you'll note that the tips (of the bullets in the magazine) get torn up over the course of firing quite often... but with little effect on accuracy.
It's all about the base of the bullet!
FMJs tend to be less uniform along the base than HPs... therefore, less accuracy. The HP may not be necessary for the accuracy, but it is there as an artifact of most HP rifle bullet creation processes.