Seeing the paragraph below I almost thought the author would move on to discuss the particular engineering horror that is logarithms of non-dimensionless values, but it is probably all for the best that this practice remains hidden.
"A related limitation, which is not always made explicit in physics texts, is that transcendental mathematical functions such as {\sin} or {\exp} should only be applied to arguments that are dimensionless"
Angles are literally dimensionless, a radian is defined as a ratio of the lengths of arc and radius.
But we all know they have 'units', or at least different formats, because there are also decimal degrees and integer DMS, but the S may also be decimal.
There are also unit formats for other ratios, e.g. decimal or %, road gradients like '1 in 4' and betting odds like '4 to 1' .
Dimensionless angles also mean that angular velocity is just 1/T. The fact of the rotation is lost (see my other comment about tracking spatial structure).
I think there is a reconciliation. If you read the other article popular today (https://news.ycombinator.com/item?id=37493955), it focuses on scale invariance as the driving principle and gives an example of units for Fourier coefficients.
I think one subtle aspect here not really covered in that other article is sort of a "duality" between "scale invariance" and "carrying scale tags" along with expressions. This also happens in differential geometry or even just analytic geometry / vectors where you can often either use an abstract geometric object notation or carry along index subscripts. So, you can think of it units as carrying along scales you care to track "for whatever reason".
The most common reason for physical units is varying conventions/devices to measure something. Angle measurers care about their own devices and so their units. By carrying along the scale you make the number a surrogate for a hypothetical experimental result. Unit conversion can be seen as how to translate from one (kind of) apparatus to another.
So, while args to trig functions are dimensionless, you do not have to be as strict about angles. You could retain them and make people carry along a multiplicative factor inside the parameters which is likely what people with instruments measuring angles in non-radians do.
Similarly, you could also (probably coherently though I have not thought deeply about it) expand SI to 8 base units adding "axial-meters". The number of base units / scales to carry along is arbitrary. It just depends upon how much expressional error checking is desired as @contravariant observes. (system conversion has its own structure as per my other comment elsethread, unelaborated by both articles.)
Because the "axialness" factor is more "kind than amount" (and binary at that), besides contravariant's (probably better) angle-dimension idea, it might be more like how `i=sqrt(-1)` creates a 2nd dimension for real numbers and you carry along that factor to make the complex plane. I have no idea how popular this kind of "complex length unit" would be in terms of error catching value for its carrying-along cost, though.
It's not as bad as you make it sound: if you write something like `e^t` where t has units of, say, time, then the right way to understand it is that it's actually `e^t (1/T)`, (1/T) is a conversion factor that is normally hidden. `ln e^t = t` then has a hidden `ln (1/T) = -ln T` factor which you can't do much with on its own, but you can carry it forward in equations and eventually if you exponentiate again, it will go back where it belongs. All of the units work out this way.
This can be useful because it allows for changing units after the fact. That `ln T` can become a `ln T(S/S) = ln T/S + ln S` if you want, and `ln T/S` can have a non-zero value that actually matters.
"A related limitation, which is not always made explicit in physics texts, is that transcendental mathematical functions such as {\sin} or {\exp} should only be applied to arguments that are dimensionless"