1. $\left[\text{corr}\left(y, \hat y\right)\right]^2$

2. The proportion of total variance that is explained by the regression model

3. The percent reduction in square loss comparing the model to a benchmark that always predicts the overall mean, $\bar y$, that is, $\left[1-\left(\dfrac{ \overset{N}{\underset{i=1}{\sum}}\left( y_i-\hat y_i \right)^2 }{ \overset{N}{\underset{i=1}{\sum}}\left( y_i-\bar y \right)^2 }\right)\right] $. This is a special case of the $R^*$ defined in equation (32) of Gneiting and Resin (2023), which they assert as a universal coefficient of determination.

"Proportion of variance explained" is a bit tricky in all but the simplest of models (OLS linear regssion evaluated in-sample), since that is those are the only times when a decomposition of the total sum of squares yielding that interpretation is guaranteed to hold. However, calculating the squared correlation between true $y$ and predicted $\hat y$ is easy enough, and calculating $\left[1-\left(\dfrac{ \overset{N}{\underset{i=1}{\sum}}\left( y_i-\hat y_i \right)^2 }{ \overset{N}{\underset{i=1}{\sum}}\left( y_i-\bar y \right)^2 }\right)\right] $ looks worse than it is, and we have a computer to do the heavy lifting, anyway.

However, these are not equivalent in all situations. That nasty fraction has the interpretation of comparing the square loss of your model to that of a naïve benchmark that always predicts the conditional mean to be the marginal/pooled/overall mean. I find it reasonable to want the data science team to be better at predicting than a middle school student would do by running AVERAGE(A:A), to use some Excel terminology, and going with that value every time. That calculation flags such situations with values below zero, indicating quite poor performance.

The predictions, at least in terms of square loss, are much worse than predicing $\bar y = 2$ every time. I give additional examples of this here and here.

Mean squared error, on the other hand, detects all deviations between true and predicted values. If the prediction is not exactly the same as its corresponding true value, there is a penalty. However, mean squared error does not make an explicit comparison to any kind of benchmark. One most certainly can say something like, "The current state of the art achieves a mean squared error of 11, so our mean squared error of 3 represents a considerable improvement," but I have seen plenty of work where all that gets reported are their results without any comparison to the performance of a benchmark or competitor, not even a naïve benchmark like, "Predict the overall mean every time."

What Pearson correlation gives you that is not as explicit in the mean squared error, however, is a sense of how tell the model is able to distinguish between distinct values. For instance, if $y=(1, 2, 3)$ and $\hat y = (11, 21, 31)$, the predictions are flawed, but the model has good ability to distinguish between distinct observed values. Pearson correlation can be seen as a measure of pure predictive discrimination, similar to how area under the receiver operator characteristic curve gives a sense of how well the two categories of a "classification" problem are distinguished from each other. Thus, Pearson correlation is not totally worthless. As you pointed out, despite the flaws of Pearson correlation, it is effective at determining if the model predicts high values when the true values are high and low values when the true values are low (ditto for Spearman correlation).

Addressig a few more pieces from the question:

This gets tricky, because there is not a universal sense of what constitutes good and bad performance that can be thought of like grades in school where a $90\%$ is an A that makes us happy and a $50\%$ is an F that makes us sad.$^{\dagger}$ I would say that the way to evaluate if a model is performing poorly is to compare it to the performance of a competitor or if the predictions are good enough to help you solve your overall task (the latter of which might be quite difficult to assess).

1. $\left[\text{corr}\left(y, \hat y\right)\right]^2$

2. The proportion of total variance that is explained by the regression model

3. The percent reduction in square loss comparing the model with a benchmark that always predicts the overall mean, $\bar y$, that is, $\left[1-\left(\dfrac{ \overset{N}{\underset{i=1}{\sum}}\left( y_i-\hat y_i \right)^2 }{ \overset{N}{\underset{i=1}{\sum}}\left( y_i-\bar y \right)^2 }\right)\right] $. This is a special case of the $R^*$ defined in equation (32) of Gneiting and Resin (2023), which they assert as a universal coefficient of determination.

"Proportion of variance explained" is a bit tricky in all but the simplest of models (OLS linear regression evaluated in-sample), because those are the only times when a decomposition of the total sum of squares yielding that interpretation is guaranteed to hold. However, calculating the squared correlation between true $y$ and predicted $\hat y$ is easy enough, and calculating $\left[1-\left(\dfrac{ \overset{N}{\underset{i=1}{\sum}}\left( y_i-\hat y_i \right)^2 }{ \overset{N}{\underset{i=1}{\sum}}\left( y_i-\bar y \right)^2 }\right)\right] $ looks worse than it is, and we have a computer to do the heavy lifting, anyway.

However, these are not equivalent in all situations. That nasty fraction has the interpretation of comparing the square loss of your model with that of a naïve benchmark that always predicts the conditional mean to be the marginal/pooled/overall mean. I find it reasonable to want the data science team to be better at predicting than a middle school student would do by running AVERAGE(A:A), to use some Excel terminology, and going with that value every time. That calculation flags such situations with values below zero, indicating quite poor performance.

The predictions, at least in terms of square loss, are much worse than predicting $\bar y = 2$ every time. I give additional examples of this here and here.

Mean squared error, on the other hand, detects all deviations between true and predicted values. If the prediction is not exactly the same as its corresponding true value, there is a penalty. However, mean squared error does not make an explicit comparison with any kind of benchmark. One most certainly can say something like "The current state of the art achieves a mean squared error of 11, so our mean squared error of 3 represents a considerable improvement," but I have seen plenty of work where all that gets reported are their results without any comparison with the performance of a benchmark or competitor, not even a naïve benchmark like, "Predict the overall mean every time."

What Pearson correlation gives you that is not as explicit in the mean squared error, however, is a sense of how well the model is able to distinguish between distinct values. For instance, if $y=(1, 2, 3)$ and $\hat y = (11, 21, 31)$, the predictions are flawed, but the model has good ability to distinguish between distinct observed values. Pearson correlation can be seen as a measure of pure predictive discrimination, similar to how area under the receiver operator characteristic curve gives a sense of how well the two categories of a "classification" problem are distinguished from each other. Thus, Pearson correlation is not totally worthless. As you pointed out, despite the flaws of Pearson correlation, it is effective at determining if the model predicts high values when the true values are high and low values when the true values are low (ditto for Spearman correlation).

Addressing a few more pieces from the question:

This gets tricky, because there is not a universal sense of what constitutes good and bad performance that can be thought of like grades in school where a $90\%$ is an A that makes us happy and a $50\%$ is an F that makes us sad.$^{\dagger}$ I would say that the way to evaluate if a model is performing poorly is to compare it with the performance of a competitor or if the predictions are good enough to help you solve your overall task (the latter of which might be quite difficult to assess).

1. $\left[\text{corr}\left(y, \hat y\right)\right]^2$

2. The proportion of total variance that is explained by the regression model

3. The percent reduction in square loss comparing the model to a benchmark that always predicts the overall mean, $\bar y$, that is, $\left[1-\left(\dfrac{ \overset{N}{\underset{i=1}{\sum}}\left( y_i-\hat y_i \right)^2 }{ \overset{N}{\underset{i=1}{\sum}}\left( y_i-\bar y \right)^2 }\right)\right] $.

$^{\dagger}$Even that is not true, though. I remember one examination in graduate school where the class average was something like a $34\%$, so when I got my test back and saw my $41\%$, I thought, "Awesome, I got an A!"

1. $\left[\text{corr}\left(y, \hat y\right)\right]^2$

2. The proportion of total variance that is explained by the regression model

3. The percent reduction in square loss comparing the model to a benchmark that always predicts the overall mean, $\bar y$, that is, $\left[1-\left(\dfrac{ \overset{N}{\underset{i=1}{\sum}}\left( y_i-\hat y_i \right)^2 }{ \overset{N}{\underset{i=1}{\sum}}\left( y_i-\bar y \right)^2 }\right)\right] $. This is a special case of the $R^*$ defined in equation (32) of Gneiting and Resin (2023), which they assert as a universal coefficient of determination.

$^{\dagger}$Even that is not true, though. I remember one examination in graduate school where the class average was something like a $34\%$, so when I got my test back and saw my $41\%$, I thought, "Awesome, I got an A!"

REFERENCE

Gneiting, Tilmann, and Johannes Resin. "Regression diagnostics meets forecast evaluation: Conditional calibration, reliability diagrams, and coefficient of determination." Electronic Journal of Statistics 17.2 (2023): 3226-3286.