But it's because the sine of 60 degrees is said by modern tables to be equal to sqrt(3) / 2, which Wildberger doesn't "believe in", he prefers to state that the square of the sine is actually 3 / 4 and that this is "more accurate".
The actual paper is at [1]:
The news from this paper (thanks for the link!) is that evidently the Babylonians preferred that, too. Surely Pythagoras would have.
But how do you actually do anything useful with this ratio ¾? Like, calculating the height of a ziggurat of a given size whose sides are 60° above the horizontal? Well, that one in particular is pretty obvious: it's just the Pythagorean theorem, which lets you do the math precisely, without any error, and then at the end you can approximate a linear result by looking up the square root of the "quadrance" in a table of square roots, which the Babylonians are already known for tabulating.
For more elaborate problems, well, Wildberger wrote the book on that. Presumably the Babylonians had books on it too.
Some tables do indeed have that value and it is a very useful value for calculation, one that can be symbolically manipulated to get you an exact number (albeit one likely expressed in radicals) for your work. When I used to teach algebra, it was a struggle to get students to let go of the decimal approximations that came out of their calculators and embrace expressions that weren’t simple decimals but were exact representations of the numbers at hand. (Then there’s really fun things like the fact that, e.g., √2 + √3 can also be written as √(5+2√6) (assuming I didn’t make an arithmetic error there)).
If you want to know how many courses of bricks your ziggurat is going to need, given that the base is 400 cubits across and there are 10 courses of bricks per cubit, you're going to have to round 2000√3/2 to an integer. You can do that with a table of squares, or you can use a decimal (or sexagesimal) fraction approximation, and I guess you're right that it isn't clear that one is necessarily better than the other.
Incidentally, the fact that we write things like 59°59'30" comes about because the Babylonians at least weren't using Wildberger's "spreads" all the time.
Personally I don't believe in either value. I prefer to state that the sine of 60 degrees is 2.7773. I believe that is more accurate.