The value of the pieces really depends on the positions, mobility, and the pieces in the right place and the right time.
Would this approach also work for estimating piece-square-tables? (Assigning each piece a value based on which of the 64 squares it is on, as frequently used by chess engines)
It should work, yes, but it's probable that amount of necessary data is going to be much more.
Imho.
Imho.