It's worth noting that the Times released this tool a decade ago. IIRC, around 2015 there was also a push for better colorblind friendly color palettes, especially on the heat map space (I remember watching a matplotlib demo, maybe, with viridis support). While there's many visualization practices we do better at now, and while this could be due for a redux, I still think it"s one of the best interactives to date. It's an OG for sure.
That's fun, and it's a much better use of heatmap since it's just a binary scale (least-most similar). When we're showing discrete options rather than a continuous "similarity" we don't want to use heatmaps because they cause undesirable blurring.
Really what the OP is trying to do is show which areas use which phrases. A heatmap could have been used where we have multiple visualizations - one for each phrase - using "Popularity" to show smooth distribution. I assume that the source data is not by county level and instead aggregated so the choropleth never would have worked great.