But that's not a web page, it's a web app! I want full control of it!

... I don't even know myself if I'm being sarcastic or not ...

Because presentation matters a lot in a lot of cases.

Your question is answered by another question. Why does more than one font exist?

Because readers have different needs? Seems pretty obvious to me.

Historically "fonts" have been set by the author, not the reader. While I appreciate that this is no longer necessary, it seems reasonable that authors still want to choose it for reasons of presentation. Of course it's easy enough to write a use stylesheet, so readers that need a different view can get one.