Why does a MP3 encoder use a fast Fourier transform before applying the psychoacoustic model?

Question

Karlheinz Brandenburg depicts a MP3 encoder like this:

Source: MP3 and AAC Explained

I marked the FFT as I'm not quite sure why it is actually necessary to perform one. Why can't the psychoacoustic model be applyed to the so called lines after the modified discrete cosine transform (MDCT) without performing a FFT?

I have some literature here, saying the frequency resolution is not accurate enough. Does this mean, dividing the original signal into 576 lines (like the filterbank and the MDCT do) is not accurate enough for the psychoacoustic model to work properly? Is the FFT more accurate?