This has been discussed in a lot of other threads, most of which I've read and a lot of the content I disagree with.
I think you have come the closest to "hitting the nail on the head" with your statement about "king volume". The interesting thing that I've noticed is that tall, slender pieces tend to look more "right" on boards with smaller squares than the "King's Base" formulas would suggest. I think there is a psychological phenomenon that has something to do with the perceived "massiveness" or "bulk" of the pieces that has not yet been explored by those who are attempting to arrive at a square size formula.
I've finally reached the conclusion that to try and reduce square size to a formula is probably counterproductive, and the only truly useful method of board square size determination is to actually try the pieces on different boards until the result "looks right" and somehow has the right "feel" to it.
I don't know where the idea came from that the ideal king for a chess board with 2.25" squares is 3.75" in height with a base of 1.75". I think there is more to consider --- like the overall size or volume of the king, as well as that of the entire chess set. In the image below, each king is 3.75" in height and from left to right have bases of 1.75", 1.625" and 1.5" , yet which looks like the biggest king? The middle one is clearly the most massive and blocks more of the board from view. The real question when determining proper piece sizing is this: what ratio of board to pieces do you want to see when viewing from a normal playing angle?