Firstly, a coordinate chart doesn't have to cover the entire spacetime and the Schwarzschild coordinate system fails to cover as much spacetime as other coordinate charts cover.
In particular, the event horizon is not part of the spacetime covered by the Schwarzschild coordinate chart.
But the books I've seen seem to treat $r$ just like the radial coordinate, and talk about "the radius of a stable circular orbit" or stuff like that.
You can get better books. The $r$ coordinate of the Schwarzschild chart is an areal coordinate, not a radial distance. And that's a pretty silly idea anyway. When a shell of matter falls towards a star/planet the distance between the shell and the distant stars increases by more than the distance between the shell and the star/planet decreases. That's life. Don't define yourself by how far you are from something, it'll bite you.
In the rest of physics, we relentlessly focus on how mathematical quantities can be measured, but I don't know how that works here, for the Schwarzschild coordinate $r$.
You can't measure $\theta$ or $\phi$ in any spherically symmetric coordinate system. So I'm not sure why this seems like a deal breaker. And you can measure the Schwarzschild areal coordinate $r$, unlike $\theta$ or $\phi$ which are unmeasurable.
In Minkowksi spacetime with coordinates $(t,x,y,z)$ you can't find the origin or any of the coordinates.
- Can the statement "the event horizon is at $r = 2GM$" be phrased in a coordinate-independent way?
It doesn't even really make sense. The coordinate chart only covers $r>2MG$ you need a different coordinate chart at the events on the horizon.
- How can the coordinate $r$ be measured?
You could measure tidal forces over a small region of time and space and compare those to the tidal forces expected for regions with different values of $r.$
But because of the equivalence principle, if you fix the precision and tolerance of your measurements and considered a small enough region, then you wouldn't be able to tell. Over a small region the tidal forces are hard to detect.
This is a fundamental principle. It's how we can actually make predictions. We state that for a small region of space and time, it's just like Minkowski spacetime (where you can't tell any coordinates, even though some directions are still clearly timelike and others clearly spacelike). You do your physics in that region, and then before you get to another region you switch to their coordinates.
The whole point of writing the metric for a practical coordinate system is to allow you to use one coordinate system in a region larger than the local freely falling inertial frame would allow.
But you shouldn't locally be able to tell. And if you look at larger regions it is essential how you bring all the information together.
But really science is about using a theory to make models and extracting predictions from the model on the one hand, and making observations on the other hand, so that in the gripping hand you can test the predictions against the observations.
Measuring a coordinate could be part of that process, but it's not what it's all about. We never even have to use that coordinate system. And we should not use that coordinate system at the event horizon.