If I recall myself correctly, cursive was taught for a few days in 2nd grade or so, and then we just wrote all the assignments in cursive. My printing is still faster, but I can still write in cursive. My signature has yet to get that adult illegibility, though, haha.

So, I think just enough to get the basics down is fine enough. I mean, it can't hurt, and it would help when reading certain fonts or older documents.

Someone upthread said that their writing in Cyrillic made their English writing better; for me, when I full integrated kana into my brain, it got that sort of round, slightly messy look that my English handwriting gets. I still try to balance my kanji, but they also show the characteristics of my English hand writing.