It is speculated that the seemingly-arbitrary number $360$ used to indicate a full revolution in degrees was chosen because the Babylonians counted in base $60$, and $60 \times 6 = 360$, making $360$ a nice even multiple of the base, which would have the same appeal as even multiples of $10$ have to us.
Gradians. Each quadrant is assigned a range of $100$ gradians. A full revolution is therefore $400$ gradians. This is more intuitive for users of a base-10 numerical system. At the time of writing, the Wikipedia article states:
Although attempts at a general introduction were made, the unit was only adopted in some countries and for specialized areas such as surveying, mining and geology.
A full revolution could be $100$ units. The number $100$, which is $10 \times 10$, would have great aesthetic appeal in our number system.
Not only would this make arithmetic involving perpendicular and opposite angles easier, it would facilitate teaching the concept of angles to primary school children because of everybody’s familiarity with providing a quantity out of $100$ such as percentages, or school grades in many countries.
Turkey switched from using an Arabic script to using a Latin script smoothly over a period of 4 years, and hasn’t looked back since. It could be argued that changing from degrees to another angle measure is a small change in comparison to changing a whole script.
Don’t we have radians?
Sure there’s also radians but we still use degrees a lot, in education for example. People who do not end up taking more advanced mathematics (most people) never learn what radians are. Also, most protractors work with degrees as opposed to radians.
So, given the argument for a switch, is there a reason that we stick to using degrees?