We do far more than train them with "mediocre handwriting". We introduce letters by showing script (which is a confusing style almost never used anywhere in the real world) and insist for years on the correct shape to use on this totally obsolete style.
Any of the activity you mention would be a better way to train manual dexterity.
This varies a lot from place to place. My school mostly focused on print letters, and didn’t provide much useful guidance to students. I agree the typical American “cursive” letters are quite terrible, and that in many places pedagogy is not very effective.
Personally I think students should learn some type of simplified italic-type script, using a pen with ink that flows well enough that they don’t need to use much pressure.
Paper furthermore remains a much more flexible and fluent tool than computer interfaces for a wide variety of tasks.