The point has been made that different people pronounce English differently. I don't think that's an insurmountable obstacle: if you made a sort of union of RP and standard American pronunciation, probably the rest of the English-speaking world would be happy to use a spelling system based on that, if they were willing to accept any change at all, that is.

A more serious problem, I think, is how the pronunciation of a single word element varies between words even for a single speaker of the standard language. How would a reformed English spelling cope with the vowels in "photograph" and "photography", for example? Perhaps there's a neat solution, but I don't know what it is. At least I'm fairly sure that phonemic transcript isn't it.