Earlier quoted context omitted.
The Palmer method descends from Spencerian script, an excessively flowery and not-all-that-legible form of writing from the days when people wanted to make their writing look difficult and fancy. Writing in a fancy style is fine for calligraphers making wedding invitations or whatever, but is a poor model for teaching children or for everyday use for most people. Italic (aka chancery cursive), a script of renaissance…
Briem is great, but can tend to very similar zig-zags when written at speed. Modern systems aim for legibility, at speed. Anyone wanting to see different systems used during 20th century might be interested in this book: http://www.amazon.co.uk/Handwriting-Twentieth-Century-Rosema...
After college, my handwriting (Zaner-Bloser, as taught in elementary school) had deteriorated into a completely illegible scrawl. I taught myself to write again, using Arrighi's Operina and Briem's commentary. If I were homeschooling my children, I'd probably use either Getty-Dubay or Barchowsky.