The tree is used to show species, too, but I can't see anyone looking at one and imagining that some weird reptile-looking thing is the pure essence of mammal. Some people are prone to getting bad ideas about ancestry (the Nazis' search for an Aryan homeland is an example) but you can hardly blame illustrations like this.
I think it's very hard to imagine what others imagine because these things aren't verbalized. It's assumptions about fundamental "how the world works" things. You can see a figure and form an assumption that you never verbalize and don't even notice that you formed an assumption.
These PIE charts do suggest a message that humanity was kinda more unified, or humans (or these regions) were still one big family at that time, all spoke PIE (again, except for faraway places), and it later fragmented.
I'm not saying that this website is bad or should be changed. I'm adding extra info, that sure it's useful, and it should be enjoyed with the knowledge that PIE was one normal language among many, existed in flux, did normal exchange of vocabulary etc with other neighboring languages, most of which are much less knowable compared to PIE. Europe used to be populated by non-Indo-European languages. In Greece, a lot of the -nthos words are likely borrowings from such a non-Indo-European language that was spoken there before Proto-Greek.
Maybe it's the curse of knowledge if you know all this. But I think the average person doesn't really have a reasonable mental map about any of this.