Could we practically forgo education in degrees in favor of radians? I've never been a fan of degrees and I'm still a bit resentful that my brain has been programmed to think in terms of them. Would it be practical to not teach them to children in primary education and just use radians from the start?
Maybe if not in the public education sector, what about something more custom like home schooling?
 A: There are pros and cons. Babylonians chose to divide the full angle in $360$ parts for the same reason they chose base-$60$ for their numeration system: $60$ and $360$ have a lot of integer divisors, so they are quite practical in defining many parts of the full angle in terms of an integer amount of degrees. That basically is also the reason for a day to have $24$ hours made by $60$ minutes, with every minute being made by $60$ seconds.
Of course, if we get rid of degrees we also get rid of a painful convertion procedure and some unhappy consequences like the fact that $\sin(1^\circ)$ is algebraic, while $\sin(1)$ is trascendental. 
But we need to give soon a solid definition of $\pi$, so we need to introduce soon the concept of length of a curve (what is a curve? How much regular do we need it to be, to give it a length?) or area enclosed by a simple closed curve (even worse).
The idea of splitting a cake into an integer amount of equal slices is probably more natural, but I am sure many of us enjoy the sadistic approach of defining $n!$ and $\Gamma(n+1)$ in pre-school, then proving the area of the unit circle is $\pi\stackrel{\text{def}}{=}\Gamma\left(\frac{1}{2}\right)^2$ and forget degrees forever :D 
A: Degrees are used in land surveying, navigation, architecture, and many other fields. It seems practical to use units that are small in those contexts.
Radians are used for the following reason:
$$
\frac d {dx} (\text{sine of }x \text{ degrees}) = (\text{cosine of } x \text{ degrees})\times \text{some constant.}
$$
Only when radians are used is the "constant" equal to $1$.
This is the same reason the number $e$ is "natural":
$$
\frac d {dx} 6^x = \big( 6^x \times \text{some constant}\big).
$$
Only if the base of the exponential function is $e$ is the "constant" equal to $1$.
