I was thinking the same thing, and I believe there are trade offs you make in both methods. If you count and group, you just have to pick your own hexcode buckets. So orange and gold are their top two colors in the example, and on our end we would jsut have to decide what range of hex values correlates to orange and what range correlates to gold and what range applies to everything. With deep learning these ranges are effectively learned, so it's more computational and feels like overkill but I can see the benefits.
If I were at work I would probably just choose my buckets and group by the hex counts though, a lot less computation in that and you can get consistent results. If I were having fun I would fit a deep net.
A company I used to work for provides custom colour classifying solutions for various industries. Classifying products into colour categories was one use case. Other uses were around perception, quantifying how different two colours were, trying to help contrast colours for colour blindness etc. They used a very different approach to this, which was a bit technical for someone not versed in colour science like me, but fascinating nonetheless. It would be interesting to compare the results between these two approaches.
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https://www.lumesca.com/