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 $60 \times 6 = 360$.
Alternatively, if $\pi$ were roughly approximated as $3$, one radian would be equal to $60$ degrees, which could be what the Babylonians did in order to arrive at the number $60$.
Gradians. Each quadrant is assigned a range of $100$ gradians. A full revolution is therefore $400$ gradians. Much more intuitive for users of a base 10 numerical system. Well-established unit of measurement used in surveying across many parts of the world.
If the Babylonians chose the number $360$ to represent a full turn because they counted in base $60$ and $6 \times 60 = 360$, it would seem logical that we choose the number $100$ to represent a full turn because $10 \times 10 = 100$.
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.
The nation of Turkey switched from using an Arabic script into using a Latin script smoothly over a period of 4 years, and haven't looked back every since. Changing from degrees to another angle measure is a small change in comparison to changing a whole script.
So, with a strong argument for a change, is there a reason that we stick to using degrees?
Sure there's also radians but we still use degrees quite a lot, in education for example. People who do not end up taking more advanced mathematics (most people) never learn what radians are. Also protractors work with degrees as opposed to radians.