The number $\pi$ is a mathematical constant that can be calculated. And indeed, as expected from naturalness, it is of order one, so it is clearly not fine-tuned. So we wouldn't be talking about fine-tuning even if it were just a physical parameter and not a mathematical constant.
Fine-tuning only occurs when a theory requires at least 2 parameters with the same units whose numerical value differs by a big ratio. That's not the case of $\pi$. It is the case of the Higgs mass in the Planck units which is $10^{-15}$, and especially cosmological constant in Planck units which is $10^{-123}$. There are many other less extreme examples.
But if your suggestion is that all such constants should finally be as calculable as $\pi$, that's indeed a classic goal of theoretical physics, one that string theory has the capacity to resolve because everything in string theory is calculable. At the current state of progress, however, there exists a very high - but finite or countable - set of possible vacua where the results of the calculations are different. That's why some people defend the anthropic reasoning.
