It was freely chosen for simplicity.

If you choose another R, the other sides (x and y) become R*cos(th) and R*sin(th)

I don't understand what is harmful about the unity circle either.

Any circle could have its radius technically be 1, as long as you set the units of measurement so that 1 equals the radius of the circle.

Because it's a unit circle.

for the same reason that rulers start with 1, it would be utterly pointless to use anything else.