Of course what is "easy" to read is always up to the individual to a certain degree. But this notation is used nowhere in math outside of logic. That should tell you something :-) Also in modern logic that is actually used in interactive proof assistants, this notation actually is not present any more, but then reintroduced when you write a paper about what you did!
Well, of course the logic notation in question is used in mostly in logic contexts. In the same manner, as a student of logic I will not come across special notation from analysis.
Although I am not working in that area any more, when I used interactive theorem provers the one I used mostly would present the proofs in Sequent calculus. It showcased a nice connection between the logical framework and a small step operational semantics reading of te formulas.
In my view, it is important to understand the reasons for different fields choices of notation; it typically reveals interesting things about the fields in question. When starting out learning a new field, it often seems strange and un-intuitive, but after a while one starts to recognize that there are many valid reasons for why a certain style of notation is used. Of course, sometimes it is just an accident of history, and bad choices stick around :)
I've hopped around quite a bit in my studies (semantics, formal methods, operations research, theoretical computer science, pure algebra, logic), and I always feel overwhelmed with new notations and concepts. Nowadays I tend to expect it when trying to learn something new.
