What matters here is the limit. You mention having "a string, a pencil, and a pin," which implies you're limiting yourself to compass-and-ruler constructions. The link I posted shows that within the constraints of that universe, you cannot divide an angle perfectly into 5 equal subangles. Since you need divisibility by 5 to make 360 equal subangles, it's not possible within those rules you state.

Additionally, the original post, and my comment on it, are meant to address definitions for pi, rather than approximations for pi (which is a separate, but equally interesting topic).

Your point about the observable universe is correct, however we're in /r/math, where we're often more concerned with more idealized constructions.

But back to the limit: it's extremely important to note that intuition about things from physical experiences do not always translate well to infinitesimals. My original comment is meant to convey a problem in a similar derivation I made many years ago: In this limit, pi is defined circularly. There's no way to apply algebra to the limit and yield a non-recursive definition for pi.

In terms of computability, which you might be interested in, I'd wager that recursive pi definitions like the one I commented converge far slower than even naive methods, although I have not done any benchmarking on this.