I gotta say.... That seems like a really good reason to me.

.... and that sounds like a cure that's worse than the disease to me. What dimensionful combination of base units could be an improvement over the trivial (that is, dimensionless) combination that the standards body has settled on?

Conversely, if it's not a combination of base units, then it would have to be a base unit in its own right, which creates a different set of problems.

Okay, so let's take a closer look at the sine expansion, which I didn't do yet:
$$\sin x = \sum \frac {d}{dx^k} \sin0 \cdot \frac{x^k}{k!}$$
That is, we're dividing by ##\operatorname{rad}^k##, and we're multiplying by ##\operatorname{rad}^k##, yielding a dimensionless quantity.
So in retrospect, even the expansion of the sine behaves perfectly reasonable with respect to radians.

In other words, radians is a good unit for dimensional analysis.

That ##(-1)^k##, is actually ##(-1)^k \text{ rad}^{-2k-1}##, which cancels (as far as the unit is concerned) with ##x^{2k+1}##.
We're just leaving out that we're making use of how radians are defined.
And I'm just saying that if something is an angle and if we express it in radians, that we should mark it with the unit radians (or alternatively with degrees, gradians, grons, or revolutions).
And if we use for instance gradians, we need to modify that -1 as well.

It does not matter if length is measured in metres or furlongs, the dimension will always be Length.
Likewise seconds, hours and seasons will always have the dimension of Time.
Speed can be mph, kph or metres/sec, whatever the units used, speed will have the dimension Length / Time.
It is pretty obvious that units carry dimensions, but that dimensions are not specific units.

Revolutions are counted be higher values than ##2 \pi##.
Radians is a derived (from a circle with radius one meter) SI unit, which doesn't mean it cannot be a dimensionless number, because it is the ratio of two lengths, and ##\frac{m}{m}=1##. It simply means, we are allowed to write ##1 \operatorname{rad}##, in contrast to e.g. ##1 \operatorname{Oz}##. It makes mathematically perfect sense to consider it as a pure number, as in

However, it can also make sense to treat it formally as a unit, e.g. when checking a physical calculation where the angles don't cancel out or in a case like this:

In the end it remains a ratio of two quantities of the same dimension, length. And as such ##1 \operatorname{rad} = 1##.

My point exactly.
We can define angles with respect to either radians, degrees, gradians, grons, or revolutions.
Still, we have to make sure to identify that unit of angle, because otherwise our calculations come to naught.