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It may make sense to just make the side-by-side view the default display. It will display side-by-side only when there is enough horizontal space, and it will display vertically when there isn't.

Well, I don't know. The table is a bit intrusive (and can become enormosuly long for large variants), and it does not really belong in the Setup section of an article. In fact it is almost obsolete by now. For side-by-side view I usually display the pieceList next to the board, rather than the piece table. This looks much more like most articles used to look, but the piece names are still clickable to summon the move diagram. It does not show the piece images, but it lists the squares they are placed, so you could look them up on the board. And when you click the name you would see the image in the move diagram. And nowadays right-clicking a piece on the board will also summon its move diagram, with its name displayed above that. So there are plenty of ways for associating the piece names with the images and the moves. You don't need a bulky table for that.

Normally I only display the piece table permanently in drop games, where it acts as the 'hand' from which you drop. Or in games with a flexible setup, where you first have to set up the board from the table.

The table would fit better in the Pieces section, but in its current form it is not really suitable for that, as it doesn't contain a verbal description of the move, just the Betza notation. The script does contain a Betza-to-text converter, which can be invoked in the Play-Test Applet to get a table suitable for pasting into the Pieces section. But it is not perfect for the more complex multi-leg moves, so you cannot rely on it for automating the article. But combined with the possibilty of a little post-editing it can be a real time safer.