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You can't trivially move pieces with magnets, what about ones that are surrounded?

It also just feels cooler to have a robot arm do it.

That's easy enogh: just put a room-temperature superconductor on the bottom of the pieces so they can start levitating!

Yeah, you would have to either 1) move those pieces away temporarily or 2) make the squares larger (or pieces narrower).

We used a robot arm to move pieces because we thought that would look the coolest. Although I agree that the over head camera isn't the prettiest solution. We thought about using RFID but we were afraid that the thickness of the board could be an issue and the RFID sensor I have found cost a lot if we are going to use 1 for each square.

These exist. https://miko.ai/products/miko-chess-grand

There you go — I thought I had seen something like that before. Open-hardware it. (Although their price does not seem unreasonable.)

https://youtu.be/AQLOj-TylGg