The simpler derivation formula was important when such symbolic computation was done by hand.
Now, except perhaps for school exercises, anything complicated is done with a computer and this advantage is much less important.
The increased accuracy and simpler formulas in other places when measuring angles in cycles a.k.a. turns vastly outweigh the advantage of radians for differentiation.
In real applications you almost always compute the derivative of sin(a*x), not of sin(x), so you have to carry a multiplicative constant through derivations anyway and the single advantage of the radian vanishes.
If you’re using a computer for symbolic algebra or whatever, none of this matters anyway. The whole post is about simpler notation for the sake of making things less error-prone when working by hand.
For a computer it matters because it is possible to compute with higher accuracy and speed the trigonometric functions with the argument in cycles instead of radians (similarly for the binary exponential and logarithm vs. the "natural" exponential and logarithm).
The main reason is that the reduction of the argument to the range where a polynomial approximation is valid becomes much simpler.
Also, the primary inputs or the final outputs of any really complete computation are never in radians, because in physical devices it is not possible to realize radians with high accuracy, but only the angles that are in a rational relationship with the cycle. This is true both for geometric angles and for the phase angles of oscillations and waves.
Now, except perhaps for school exercises, anything complicated is done with a computer and this advantage is much less important.
The increased accuracy and simpler formulas in other places when measuring angles in cycles a.k.a. turns vastly outweigh the advantage of radians for differentiation.
In real applications you almost always compute the derivative of sin(a*x), not of sin(x), so you have to carry a multiplicative constant through derivations anyway and the single advantage of the radian vanishes.