The principal value of $\arctan x$ is between $-\frac\pi 2$ and $\frac\pi 2$. But then why isn't the principal value of $\operatorname*{arccot} x$ the same as that of $\arctan x$?
I know that this is only a convention. But in every other cases of principal values of inverse trigonometric functions, I can find a pattern
The modulus of the elements of the range of inverse trigonometric functions are always kept as low as possible.
If there are two different ways to determine the set principal values, then the one consisting of the positive portion will be preferred.
This holds true for every inverse trigonometric function except $\operatorname*{arccot}x$. So, then why does $\operatorname*{arccot} x$ not comply?
There are two other desirable properties for the inverse function --- continuity and monotonicity. For $\sin$,$\cos$ and $\tan$ they are in agreement with the other guidelines which you propose. However, if you take the inverse of $\cot$ on $[-\pi/2,0)\cup(0,\pi/2)$ (the disconnectedness is another immediate drawback), then you get a nasty jump discontinuity at $0$.