It's the right way to think about the structures, sure. But when you want to look up the laws for what makes something a Ring, for instance, having them all in 1 place instead of having to jump between: Monoid, Semigroup, Group, Abelian Group, Quasiring, Nearring, and Ring. (And some of those laws occur twice, for different operators) makes it a lot more useful as a reference document.
That's like saying when implementing a function you shouldn't call another fn, you should copy and paste its implementation so you don't have to flip to another location in the code.
The right way is to learn what the function means so you can set see calls to it and know what it does.
If the page had arithmetic structures, then obviously yes. Groups don’t include those properties, which is why they are not listed, and that is a very important distinction.
