Or rather, perhaps both the symbols and positions for x and y should be switched. The convention of writing coordinates as (x, y) is so deeply ingrained that, in my opinion, it causes confusion in situations where x doesn’t naturally come first.

Reasons for x first

As I see it, the primary reason for writing x first is the function relationship, i.e. the dependence of y on x. We think of an input as existing first, then “producing” an output. That being said, note that the slope-intercept and point-slope forms of a linear equation start with y, characterizing the relationship more like “y comes from x” than “x produces y.”

Reasons for y first

The first situation where x first can be confusing is slope calculation. Recall that the slope of a line is given by the change in y divided by the change in x, or “rise over run.” A common mistake is calculating slope as change in x divided by change in y. Intuitively, since x “always” comes first, it seems to make more sense for it to go in the numerator.

The second situation where x first can be confusing is in the unit circle. Points on the unit circle have coordinates that can be defined by trigonometric functions of the central angle. Recall that the x-coordinate is the cosine of the angle and the y-coordinate is the sine of the angle. Usually when we talk about sine and cosine, sine comes first, but in coordinates on the unit circle it’s cosine followed by sine. Again, mixing up x and y on the unit circle is a common mistake and I think it’s because we usually think of x and sine as both coming first.

It’s too late

Unfortunately, such a change is basically impossible. The convention is too well established and attempting to change it would be too confusing. It goes to show, however, that mathematical notation is a human invention and always could have been different. The way we happen to do things is not necessarily the best way. In general, notational decisions always have trade-offs. Something that makes the most sense in one situation may make very little sense in another situation.