We need to introduce these simple demonstrations in math courses.
They aren’t used later on because, for more complicated expressions, manipulating the algebraic formulas is easier than the geometric method, or completely breaks down (e.g. when the powers aren’t integers or are variables)
For example, I think anybody who can visualize what (a+2b)³ or (a+b)⁴ looks like geometrically can also, and easier, do the expansion algebraically.
No, or at least they were not in the high-school math books I was assigned. There were no explicit referrals in any way to visual representations of the algebra involved.
It doesn't mean it is not possible to have rigorous visual proofs, the one in the article is, but doing so can be deceptively hard. It is also common for visual proofs to be less complete, for instance, here, it only covers the cases where a>0, b>0 and a>b, there are no such limitations when doing so with algebra. I guess you can tweak the visual proof to account for these cases, but it will become far less elegant.
So I understand why teachers avoid visual proofs in math classes, they actually want you to stay away from them. They have entertainment value, that's why you see them so much in pop-science, and I think it is important, and also some historical value, as they were much more present in ancient times, but to actually learn maths, not so much. That is unless you are studying visual proofs specifically, but I think it is an advanced topic you would tackle well after you can master simple equations like a^2 – b^2 = (a + b)(a – b).
a ( b+c) = ab + ac.
I agree it's important that students intuitively feel that
a ( b+c) = ab + ac
is true.
But once we are at that, why would we even bother to teach or "proof" all variations of that expression?