Another thing I've encountered: in South India, there is a measure of distance & time called a "nazhika". It is 24 minutes. 2 1/2 nazhika is one hour, 60 nazhikas make a day. A nazhika is also roughly the time taken for a normal person to walk 1 mile. Hence a nazhika is also used as approximately one mile. Seems a bit weird until you remember that light-years involve the same identification of time and distance.
Non-metric units like these and others ("foot") have probably very good reasons behind them, and they could be used by people in their day-to-day activities while not carrying measuring instruments with them. Metric System is more systematic, but people "lose touch" with intuitions of quantity.
The Sumerians did, in fact, periodically insert intercalary months in order to offset the difference, which suggests that this is in part close approximation and on the other part nice for application.
Of course, at some point it becomes a huge pain and you buckle down and choose something just as arbitrary but easier to handle. And that's why we don't count angles in the milliseconds since 1970. Or days in a year; though we do sometimes measure 3D angles in solid minutes, which seems like a metaphor taken too far :)
Even in metric-using countries, nobody uses it for time.
For actual applications of spherical trig though degree, minute, second makes a lot of sense, primarily because the earth rotates about one arc second every 4 minutes, and if you know this then you can do manual navigation via the stars and many other things.
However for many things I do use metric time, just not for the human aspect of it. For example, for one customer (admittedly in the sciences), we had to help them estimate how much hardware they needed for additional load. So you do the work in seconds because at that point the math is easiest, and convert to ratios following.
Of course for practical use using SI second wouldn't be very good solution. Traditionally second is derived from the length of day, and I think that would make sense for metric time too. 1 milliday would be somewhat close to 1 minute and 50 millidays (or maybe half deciday) would be close to one hour. Of course the name probably should be something else than "day" to reduce confusion.
So I could be working something like 16 decidays next week.
The number of sides depends on the divisiblity of 360 by the number of angles so:
3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 30, and so forth.
For angle measurement where you want a closed geometry figure at the end, degrees are extremely elegant, and 360 is 2^3x3^2x5
You'd also lose the small-angle approximations sin(theta)=tan(theta)=theta which in my field are used extensively to convert nonlinear to linear equations.
Nonetheless, your proposed unit already exists and is called tau. Or write it as 2pi if you want to be more easily understood.
And when I say natural, I don't mean elegant. They often tend to be thoroughly arbitrary. But they match the needs, which is usually reasonably pragmatic. For example, 360 degrees is ideal for simple in-head directional geometry, but radians are by far simpler in algebra or very precise measurements (because Pi can often be neatly factored out).
Once you start using enough decimal points, all units are lousy. I remember using angstroms in astronomy because it fit better into the optics theory and the distances are already absurd it didn't matter. So you may as well use the units that are convenient, and just get good swapping.
(PS: glass sheets are sold by the square foot, but in thicknesses measured in millimeters. Turns out to be pretty convenient that way.)