One reason for starting with the first half of a natural turn is that you will rarely get as clean an opportunity to do another inline one during the dance.
There are ways to set one up of course, but they each bring additional challenges which are likely to have a negative impact on the quality of the natural. For example, it's drastically harder to take a perfectly swung natural after a closed change than after a prep step (you will almost never see it in open competition), but the fact that it's harder makes it a great thing to practice.
And of course there are many ways to get into an outside partner natural, but that's not exactly the same figure.
Starting off with the classic figure done your best is important to competitors who want to make a good impression on judges, but it can also be nice at a social where you make to make a good impression on a partner.
Some of the other entries you suggested can be fun to work on, and might be good choices if you find yourself in a a situation where a natural won't fit, for example at a crowded social or if someone has tried to put an unreasonable number of figures on the floor for a the first round of a competition. But no one, world champs included, is going to execute them in a way that can compete with what the classic natural opening has to offer when space permits.
Most of the other dances are a bit more welcoming of alternate beginnings. Though the initial feather in foxtrot is almost as traditional, it's not unusually to see someone begin with something else. For beginners tango might typically start with two walks and quickstep with a natural, but there are all sorts of common choices in those later on.