This operates on the assumption that eye tracking and brain waves signals are positively or negatively correlated (for which the setup you propose can be thought of as an estimator with control variates). Is this indeed the case?
Eye movements cause travelling waves to sweep across visual brain areas (which are actually a huge chunk of the brain), potentially swamping whatever was going on before. Some of this is probably supports visual processing (by, e.g., persisting or resetting neural activity across eye movements), and some of it might be epi-phenomenal.
Furthermore, EEG electrodes pick up huge, non-neural signals from the movement of the ocular muscles and the movement of the eyeball itself. This can be reduced with clever pre-processing and more expensive amplifiers (e.g., that don't saturate or recover from saturation more quickly).