Using a keyhole contour I've noticed that some complex analysis textbooks discuss evaluating real-valued integrals like $\int_{0}^{\infty} \frac{\sqrt{x}}{1+x^{2}} \, dx $ using a keyhole contour before they have defined the Cauchy principal value of an integral (first definition). But isn't using a keyhole contour a principal value approach in the sense that the contour approaches the singularity at the origin in a symmetrical way?

The reason to use a keyhole contour is to do with the fact that $z^{1/2}$ has a branch point at the origin.

You don't really need to know the concept of Cauchy Principal Value to be able to take the limit of the inner radius going to $0$.

But, in a way you are right that it is a principal value approach.