Yes, allow me to elaborate.
Recall that for some outcome $y_i \in \mathbb{R}, \forall i=1,2,..,n$ we define MSE and $\textrm{R}^2$ as
\begin{equation}
\textrm{MSE}(y, \hat{y} ) = \frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y_i})^{2}
\end{equation}
\begin{equation}
\textrm{R}^2(y, \hat{y} ) = 1 - \frac{\sum_{i=1}^{n} (y_i - \hat{y_i})^{2}}{ \sum_{i=1}^{n} (y_i - \bar{y})^{2} }
\end{equation}
So, as you noted, $\textrm{R}^2$ is a normalized version of MSE, we use MSE for reporting because I think it's a simple metric and it is technically the loss-function we are minimizing when we solve the normal equations.
$\textrm{R}^2$ is useful because it is often easier to interpret since it doesn't depend on the scale of the data.
As a concrete example, consider two models: one predicting income and the other predicting age, $\textrm{R}^2$ will make it easier to state which model is performing better.*
*In general, this isn't a great idea and you shouldn't compare metrics like $\textrm{R}^2$ across different models to make these sorts of claims because some things are just fundamentally harder to predict than others (e.g., stock markets vs. who survived the Titanic).
