$$\text{var}(X)=\text{E}(X^2)−(\text{E}(X))^2$$
I am able to follow the proof of the above from the basic definition of the variance, and mathematically it seems fine but I cannot seem to grasp intuitively why it is true.
Surely such a simple formula must hold some deeper significance or a clear, intuitive explanation?
As I think you point out, $var \ X = E(\,(X - E(X))^2\,)$, a centered second moment of the distribution $X$. This version is full of intuitive content for me.
Even so, if $E(X) = 0$, then $$var \ X = E(X^2)$$ and thus we could argue the $-E(X)^2$ term is a correction to recenter the moment.
Sometimes in mathematics, we have to be content with the fact that one form of an expression has more intuitive content for an us than an equivalent form. I would argue that's part of the power of mathematics.
Another example is the formula for the Laplace transform, which is not especially intuitive but with manipulation can be seen to be a generalization of power series, which is intuitively richer.