It wasn't a problem with people getting sick of video games. It was a problem with Americans getting sick of the Atari 2600, other similar video game consoles and the distribution model of cartridges sold in toy stores
The sales of home computers in the US massively increased during the "video game crash", especially sales of the Commodore 64. I suspect those C64s spent the majority of their time playing video games, off floppy disks.
Nintendo must have done their market research, because they clearly identified the source of the problem and cleverly marketed the NES as "Not a video game console" and "Completely different from the Atari 2600".