As others said, degrees are taught, since they are still used. So, the question becomes why are they still used.
To purely work with fractions would not be very convenient for various somewhat everyday things, since many people are more used to/better at operating with integers. So to really use $\tau$ and fractions thereof seems incovenient, and one somehow will want to rescale.
And, one could say, this is actually how degrees arise, one rescales by a factor of $360$.
Now the question becomes, why $360$ and not something else, such as perhaps $100$.
The first answer is that some do rescale by something else, there is another unit called gradian where a right angle has $100$ gradian, which came up along with the metric system. There is (are in fact) also mil and a couple other things, see the Wikipedia page on angle.
The second answer is that $360$ is quite reasonable, more so than things more based on $10$, if one is interested in geometry.
To construct a hexagon is an extremely natural thing, so obviously a sixth of a turn is important and should be 'nice.' And a right angle being important too, this should also be 'nice.' So aready the scale should be divisible by $12$. Now, throw in the mix that also the constrcution of a regular pentagon is known since ancient times, and you are at multiples of $60$.