I really don't appreciate the argument that "it doesn't matter what the constant is". Of course it doesn't matter if you are only interested in the result of a calculation, but like you say, it does matter when you're trying to understand what the formulas express.

I find it surprising that this article, and many mathematicians asked to comment in the media, pull out the area formula as their prime counter-example, seemingly forgetting that using pi is only obscuring the mechanics behind it.