Can a NFC tag be read over a large flat area
This is quite tricky - NFC is not designed to do this, and it's not really designed to cope with lots of readers in a small area either. If you did go for one per square, you'd have to program them to make sure that only one was active at any one time, and then cycle rapidly through all of them. The final result might be quite slow.

"Traditional" chess computers avoid all of this by not actually identifying the pieces individually. They know the starting layout, and they can tell which squares have a piece on them (contact or IR-reflective proximity sensors). They then watch for a "piece removed then piece placed" sequence and assume it's the same piece.

There are various tools like the Apple Pencil which provide very precise identification of where they are on a capacitative touchscreen. But they work by emitting a signal themselves, so require charging, and active electronics in 64 chess pieces is likely to be a huge pain.

You could perhaps borrow something from the Tritiled project and put an extremely low power microprocessor, battery and LED in each piece, and then have it emit an extremely short, dim, pulse-coded flash downwards into the board where sensors could read it. Best case battery life would be a few years from a coin cell.

Or - as has just occured to me - put a coil in the board and each piece, and use that to power all the individual identification lights.