This particular problem pops up in quite a few domains. (We often refer to it as "phase unwrapping" in the geosciences.) The approach here is a good one so long as your noise doesn't result in lots of mistaken "wrap-arounds".

However, it will fail badly in the presence of noise in many cases. It's particularly problematic when the slope changes (e.g. a polynomial) or where the slope is high and the noise is high. (Note that polynomials are still linear in the sense mentioned here: linear regression).

At any rate, this is definitely a nice write-up, but a bit more discussion of where the approach breaks down would be useful. It's actually a classic example of an elegant solution that breaks down frequently in practice (i.e. it's commonly used as a teaching example in various courses). A better solution is usually more complex, domain specific, and therefore out-of-scope, but failure modes for this method make for a nice set of examples.