I'm not disagreeing with you Scott. At all. I personally don't know anything about taxonomy of algae. But you got me curious, so I googled it. Tis amazing the variety of opinions, and it seems there has been some change in classification. I tried to go to more scholarly sites as I have told my kids many times not to be citing Wikipedia to me.

So it seems the current thought is algae are truly not plants. Though they admit that they are the predecessors of plants and I can't quite figure out how a predecessor is not in the kingdom. Then I went to TAMU.
http://aquaplant.tamu.edu/plant-identification/alphabetical-index/planktonic-algae/
"Planktonic algae are floating microscope plants that are normal and essential inhabitants of sunlit surface waters."