The tree appearance just makes it too inviting to imagine PIE as some pristine ur-language, the kind of original language of Babel. I'm not saying that people explicitly think these thoughts, just that it's easy to get this impression, even if it's not conscious. The fascination about the original human tongue in Eden used to be a common intellectual preoccupation. My point here is that PIE is quite a different story.
Is it though? It's not as if it can pull in hypothetical superfamilies like Nostratic. It's not saying it's a family tree of all languages, just I-E. Your assumption is that somebody would look at it and forget that Mandarin exists, or Arabic, or Finnish, &c. I don't buy that. Hence my fruit joke. I'm not sure exactly what you expect them to do to avoid an egregious misreading by somebody spectacularly smoothbrained, because anyone with that mentality isn't going to be paying attention anyway.
[Removed my Anatolian criticism: I just hadn't zoomed in enough!]
No, I understand that people would know that Chinese existed far away. My point is that locally, PIE had neighbors, had variants, dialects, it had earlier forms, it was like a living language in a normal human context.
Maybe your brain works differently. I had this kind of misconception, imagining languages to be Pangea-like. That there used to be less linguistic diversity, there was a big uniform language (and perhaps a handful of others far away on the Earth) that later on fractured. But reality is more dynamic. PIE was just an ordinary language. And there was likely more linguistic diversity than today, due to less communication and travel opportunities and geographical features being more isolating.
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, 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.