Does Stockfish really use a Leela-inspired NN? I thought the NNUE was independently developed and completely different (it's a very tiny network that runs on the CPU).
To get the evaluation of the whole position, you add up all of these mappings for the pieces on the board with opposite signs for the opposing players.
If you blur your eyes a little, this already looks a lot like a neural net. It's just a big summation of terms, and if you leave in a 0*(whatever value) for every piece that's not present, you've effectively embedded your lookup table into a giant mathematical expression that can be optimised by gradient descent.
The reason computer shogi programmers stumbled on this is that they were experimenting with adding more dimensions to the piece-square table, specifically via indexing by king position as well. So now you have 4 or 5 dimensions, making for a pretty massive array. Hand-tuning all the values becomes less and less feasible, and so I think discovering this idea of rearchitecting it as a neural-net was more or less inevitable.
So NNUE is actually just a pretty natural evolution of what they were doing before.