Are there any statistical properties from training data that imply whether feature selection will help improve an ML model's performance? I was just wondering if there are any statistical tests/values we can look over the training data to know if applying feature selection can improve a model's performance when training on the data (assuming testing data is distributed similarly)?

As I show in Chapter 4 of RMS you can use the Efron-Gong optimism bootstrap to get an overfitting-corrected estimate of model performance with and without feature selection.  For either case you can use penalization to improve estimates, but you will typically find that the attempt to be parsimonious through the use of feature selection results in worse models, as compared with the best available full modeling approach (an example of the latter is ridge regression).
