Hi there,
you could do something similar using the WP Show Posts plugin:
https://en-gb.wordpress.org/plugins/wp-show-posts/
and GenerateBlocks:
https://en-gb.wordpress.org/plugins/generateblocks/
Use WPSP to create a list for each category. Setting it to 1 column and just the title.
Then use the GenerateBlocks plugin to create your grid.
In each grid item you can add a Headline, the relevant WPSP Shortcode and a Button linking to the main archive.