Isn't this focus on the fractal nature of a measurement based on sums of ever-more-microscopic measurements is a bit misleading?

Maybe we should turn the question around. Is there a measurement of altitude gain that is clearly wrong? And I think that's what O.P.'s collection is. The wide, 20% of mean standard deviation, variation of climbs over a half-dozen or so repetitions indicates that the measurement is unstable with the given cell-phone GPS instrument. Terry Morse's Strava case is another bad measurement example.

If we start from the premise that we need to identify and correct bad measurements, surely noisy cell phone GPS tracks are one of the very first things to eliminate from consideration. If someone, for whatever reason, wants to track climbing and is having problems running a cell phone tracker, one of the first responses is "cell phone GPS trackers are often unreliable." A simple corollary would be, "get something else to track climbs on your typical route."

I would be somewhat interested in why the O.P.'s route had such divergence. Did the route go through a downtown or near rock faces that would generate spurious points on the GPS track? Did the phone lose track because of tree leaves? Or was it an example of riding in the left tire track generates a different profile than riding on the shoulder?