You haven't answered my question at all. Why is it logical or reasonable to temporarily skip the rest of a range and start off somewhere else, when it is exactly the same thing (from the point of view of probabilities of a uniform distribution) to continue scanning the range?

Let's say you scanned 1/3 of the range and by your calculus you decide that you should skip to the key at 2/3 of the range. However, the distribution behaves uniformly, so whether you scan the first and last third, or the first and second thirds, have identical probabilities, and also identical cumulative probabilities, no matter how you look at them. Why would one choose to take the effort to go at 2/3 instead of simply continuing from 1/3 + 1? Except boredom and excitement, not much reason to do that.

Maybe someone would think that if they move around the space they have higher chances of spotting the key. This is an illusion. The number of random picks until the lucky winner is hit is, on average, exactly the same as if the range is traversed from some whatever point continuously, on average.

You tried to prove something that makes absolute zero sense in the context of the puzzles (which is to find a key in a dataset that behaves like a uniform distribution, though it is created via fixed rules), not to play around with post-factum explorations), and you've been told your strategy can only work and make sense once you actually know all the real probabilities before-hand.

But to have those numbers, you have to scan all the keys, which defeats the purpose, since you'll find the correct key along the way. In which case I don't think it would matter at all what's going on the inner guts of the range.