The problem is that SATs, and therefore schools, prioritise the use of cursive, but not the teaching of cursive.
When I was taught cursive in the 70s we were required to sit tucked in at our tables and we always wrote in lined notebooks positioned at a slight angle in front of us. Before we even got to that stage we were explicitly taught how to hold our pencils and corrected/supported to change our grip.
In the 21st century all my children were taught handwriting seated cross-legged on the floor, writing on plain whiteboards on their laps. 2-3 times a week they got 10min handwriting practice in lined notebooks at their tables. If there was time for it. No mention of how the books should be positioned. No explicit teaching how to hold their pencils. The tables were often hexagonal, or, if square, with children working on both sides of the corner, so there was no space to angle their books. How can children be expected to learn clear writing under these conditions? At primary school children write nearly as much as they did 50y ago. There is still the opportunity to write and establish a clear hand.
And don't get me started on this ghastly version of precursive they now use. Instead of every letter leading out, they all must now lead in as well. So the children are taught to begin and finish every letter on the writing line. Fair enough. Until you have to write something like frog. What they end up writing is fnag. Because they then have to unlearn r and o.
When my left handed dc was in KS1, I kept explaining to the teachers that he needed to angle his work to the right, and at a steeper angle than right handers do. It made no difference. They even made excuse that there wasn't space at the tables to allow him to angle his book! By end KS2 his writing was entrenched, and the stress of the school finally trying to change it (for the SATs!) was too much. Unsurprisingly, ds's handwriting is illegible. So much so, that he has to do his GCSEs on a laptop.