Does PCA provide advantage if all PC's are used? The question is basically stated in the title. Does principle component analysis even provide an advantage if all principle components are used later? In an application I am looking at PCA is conducted before a simple regression. The author claims that if all PC's are included in the regression, the result will be equivalent to OLS on the original feature set. Is this true?

The PCs are just linear combinations of the original features. For example, if there are two features, $x$ and $y$, the features mapped on the PCs will be something like $f_1=\alpha_1 x+\beta_1 y$, and $f_2=\alpha_2x+\beta_2y$. So, it's just a change of axes.
In ordinary linear regression, the target variable is expressed in terms of linear combination of features, i.e. $y=ax+by+k$. Using the new features that are linear combinations of the old ones will generate an equivalent equation. For example, for two features, this would look like the following:$$\begin{align}y&=cf_1+df_2+k=c(\alpha_1x+\beta_1y)+d(\alpha_2x+\beta_2y)+k\\&=\underbrace{(c\alpha_1+d\alpha_2)}_ax + \underbrace{(c\beta_1+d\beta_2)}_by+k\end{align}$$
This is the case for OLS, but in general, does using all PCs have an advantage? Maybe. Having orthogonal axes may be paramount for the downstream analyses you'll perform depending on what you’re after, so generalizing this for all ML is not possible.
