Because it makes math solely about memorization and doesn't involve number sense (discussed in the original article, too), and causes anxiety for those students who struggle to shallowly memorize the table. See also work on 'growth mindset' vs. 'fixed mindset'. When or if students make mistakes at memorizing, they might simply give up and think they just can't do math.
I don't think it's an either/or - you can introduce the tables, but do so in funner or more subtle ways. As an example of the latter, I was in a combined 2nd & 3rd grade class. We weren't supposed to learn multiplication until 3rd grade. There was a huge times table on one of the walls that I'd walk by several times a day all year in 2nd grade. Gradually I'd get curious about the patterns I saw in the table. I never 'memorized' it (until 3rd grade when we had to), but I felt like I had begun to understand it and how it 'worked' after repeated exposure to it.
But I like the idea of the original author to make games about it. Another idea is to have students create their own tables.