I think anxiety is the key here. Everyone is capable of deliberate practice, but the degree to which one can sustain it for an extended, discrete session varies

I think anxiety is the key here. Everyone is capable of deliberate practice, but the degree to which one can sustain it for an extended, discrete session varies a lot by how the stress people feel from outside sources or from the process of mistake-making inherent in practice intrudes on and interrupts their focus. It could be that some subset of average players would be better if the stressors could be dealt with.