Does the brain stop processing data when the eyes move?
I'm sorry you happened upon that research from the 70s. Should it have been true those with Nystagmus (constant involuntary eye movement) would never learn anything so it must be in a general sense false.
Here is a current thoughts and research into the subject:

Eye movement-related brain activity during perceptual and cognitive processing
For several decades researchers have been recording electrical brain activity associated with eye movements in attempt to understand their neural mechanisms. However, recent advances in eye-tracking technology have allowed researchers to use eye movements as the means of segmenting the ongoing brain activity into episodes relevant to cognitive processes in scene perception, reading, and visual search. This opened doors to uncovering the active and dynamic neural mechanisms underlying perception, attention and memory in naturalistic conditions. The present eBook contains a representative collection of studies from various fields of visual neuroscience that use this cutting edge approach of combining eye movements and neural activity.
The majority of the articles in the eBook combine the measurement of eye movements with the recording of the electroencephalogram (EEG) in human subjects performing various psychological tasks. The most common methodological approach is examination of the EEG activity time-aligned to certain eye movement events, such as the onset of a fixation or a start of a saccadic eye movement (Fischer et al., 2013; Frey et al., 2013; Henderson et al., 2013; Hutzler et al., 2013; Nikolaev et al., 2013; Richards, 2013; Simola et al., 2013). Several works employ the time-frequency and synchrony analysis (Fischer et al., 2013; Hoffman et al., 2013; Ito et al., 2013; Nakatani and Van Leeuwen, 2013; Nakatani et al., 2013).