Left-to-right writing systems are optimized for right-hand use. Two examples:

* if you're left-handed, your hand smudges over the ink before it dries. There are various contortions that some left-handed people do (hover the hand or wrap it around from above) - right handed ones don't need any of that.

* stroke patterns, as usually learnt in school, result in pushing away if left handed, vs drawing to, if you're right handed. This results in less ideal strokes, and if you're working with a sharp pencil/pen on a sensitive paper, this can tear the paper. If you're working with a felt-tip pen, the line width/pressure suffers as well.

That said, if you really make an effort, you can have a pretty decent handwriting if you're left handed. And if you are forced to use right hand when learning handwriting, you can still have a pretty decent handwriting.

I'm not familiar with details of chinese handwriting (what's easier/better if you're left vs right handed), wouldn't be surprise the constraints are similar.

So I guess your remark about messy handwriting is related to the strict standards for the students (which includes expectation they must write with right hand).