Is there a physical argument for the Kutta-Joukowski theorem?
The question as asked in the title is one of the great debates of the discipline of aerodynamics (and you can see by the number of times I've edited this answer that it's still bouncing around in my own head). If you've gotten this far in Anderson and are making this kind of inquiry, you must read McClean. The first 300-or-so pages are devoted basically to answering this very question. Basically: Yes, there is a physical argument (see Peter Kämpf's answer and Anderson 4.5), but saying that lift is caused by circulation is too much of a stretch (which Anderson notes in section 3.16).

Let me turn to your specific example.
Unfortunately, I think Anderson is trying to force the mathematics into the physics rather than start with the physics and describe why the mathematics is a good representation (as McClean sets out to do from the beginning). "Make this physical argument by drawing a closed curve around the body..." basically tells us to make a physical argument by first using a mathematical construction, which to me doesn't actually shed any light on the physics. Further, then we're supposed to be thinking about infinite distances (not physically useful), a "speck on a page" (which has nothing to do with airflow), and really just make use of $\Gamma$ (which is defined purely mathematically) rather than describe a physical flowfield. In short, the closest thing to a "physical" argument here is saying more or less that any two things look the same when viewed from far enough away. This explanation does not seem convincing to me. If you put lots of little vortices around the profile of a cow then reduce it to a "speck on the page," will the cow generate the same lift as an airfoil? To get at the physical generation of lift, we need to remain close to the lifting bodies and determine why their specific geometries create flowfields that produce lift. (Consider, for example, why we need a Kutta condition on an airfoil but not on a cylinder.)

Actually, the mathematical proof (based on conformal mapping) uses the same basic "physical" argument that Anderson makes. Conformal mapping tends to be more of a graduate-level topic, though, so beware of feeling like you need to understand all the underlying math at this point.

In summary, Anderson does not do a good job here of explaining physically why the airfoil has the same lift as the cylinder. If you truly want to understand the physics, read McClean. Anderson's "physical" argument is really just a boiled-down mathematical one.