alt Hacker NewsIt is incorrect. They are not plants. I am curious why you think it is correct to call protists plants? It is not just taxonomy, but ecology, life history, tissue structures, and so much more. Some plants do photosynthesis. Some algae do photosynthesis. Not all things that do photosynthesis are plants. Not all things that do not do photosynthesis are not plants.
Presumably because "plant" was a word used to describe all sessile, photosynthesizing organisms visible with the naked eye long before we developed sophisticated enough tools to scientifically separate sea-weed from other plants.
Also, I'm not an expert, and I know Wikipedia isn't the most authoritative source, but https://en.wikipedia.org/wiki/Plant#Alternative_concepts seems to suggest that green algae are, at least at times, scientifically considered the parent group of land plants. So there's that, too.