Does a similar system to a pangram exist for handwriting practice
I think you've pretty much answered your question. No there is no exact equivalent - an easily-memorised group of phrases that you can write repeatedly, to practice handwriting. There are two options:


For basic practice of writing strokes, Eight Principles of Yong is fine; you may even practice one stroke repeatedly, for example if you were doing brush calligraphy
To practice getting the shape and positioning of strokes/characters right, it is necessary to practice with a large corpus. Thousand Character Classic is a good option but you can substitute with your favourite poem or article. The point is to practice writing strokes and radicals that are different shapes and sizes, and getting their proportions right.