Vorpal
Neither a dandy nor a clown
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That's an odd way of putting it, especially given the Galilean option, while in STR any timelike direction is as good as any other. What distinguished the Euclidean one is is inconsistent with causality due to not distinguishing space and time, which is I think what you're trying to get at. This is a question allowing a consistent ordering of events rather than a preferred direction. You can turn around in Euclidean space, but not in time.Eventually you'll get down to geometry. And you can ask 'why' for that, again, but this time there's a decent answer: "While it didn't have to work that way, having a time dimension that's just like a space dimension would mean there's no preferred direction of time. For (complicated reasons), this makes life unlikely to ever appear."
But you're correct in that there is a sense in which there are three simplest geometries: {Euclidean, Galilean, Minkowski}, of which the first is inconsistent with causality while the second has a degenerate metric: temporal and spatial measurements are disconnected from each other. So in a way the STR case is the most well-behaved geometry out of the three simplest cases. One technical characterisation of the specific sense in which those are simplest geometries is: isotropy of space, homogeneity of time, and inertial frames forming a group.