As to the "the case" that "80% of the population of the Spartan state were slaves", that is an acceptable turn of phrase, but it is not what the blog author says. He says that the majority of the people of "Sparta" were slaves. This is calculated to come across as a tear down of the myth of "noble" Sparta because it makes no distinction between the free citizens of Sparta who are the ones elevated to mythical status by ignorance of history, and their slaves, who are never and under no circimstances considered to be part of the "noble warriors of Sparta" or what have you.
In short, the author is making a very clear attempt to exploit the confusion caused by the synecdochical use of "Sparta" to mean both the city and the city-state, to make a point that ultimately only serves his vanity, and is not really informative. To say that "most people of the city state of Sparta were slaves" is only revelatory to those who have no idea of ancient history.
To give you an analogy, it's like someone making a huge todo about how "Black holes are not really black". This can only be news to people who don't know what black holes are and what is meant by "black" in their name. Then instead of trying to clarify the meaning of the word "black" in "black hole" the author launches into a tirade about how black holes have been glorified as being the blackest bodies in the universe when in reality they are not even black and most of their mass is invisible because it's hidden behind the event horizon and Oh Gods! All you 've ever been told about physics, it's all wrong! You've been worshipping a black body that is not even black and is not even all there!
It's a lot of noise for nothing but internet likes and an utter appeal to ignorance that does not improve the reader's knowledge one bit. And it is many peoples' first point of entry into the subject of not only Sparta, but ancient history in general. If that first point of entry is not 300 instead, which to be sure is even worse but without which the author's blog post would not even exist, because it has nothing to say other than "300 sucks, man".
300 has Persians throwing incendiary grenades. Their king is a giant. He's carried to battle on a golden throne, like a Warhammer figure. But the author saw fit to attack 300's realism by picking apart its mythologising of "noble" Spartan warriors, which nobody has ever believed other than people who learn their history from comics and movies. This is how much the author of the blog post is interested in historic truth.