For me there are 3 problems with SATs.
First is the narrowing of the curriculum and educational experiences in Year 6 - it has become a boring and stressful year for children and no one wants to teach Year 6.
It's all SATs prep until May, and then nothing for 4 months until children start secondary in Year 7 so there's a huge amount of catch up to do.
Second is the amount of stress that children are put under - schools pay lip service to the idea that it's not a test of the child, it's the school, and this test doesn't define you. But then they subject children to revision sessions, breakfast clubs, mock tests, target children on grade boundaries.
Third is that SATs results are used to predict GCSE grades. Doing well in your SATs is a rubbish indication that you'll be good at GCSE art or languages. Children who spent Year 6 being coached and target to push up their SATs results then spent the whole of their secondary education being told they are failing and under target because they are heading for Bs instead of the predicted As.
So, to improve things I would move the test to Term 6 and ban all test prep before the May half term holiday. I would ban schools from offering or requiring children to attend SATs revision beyond the normal school day. I'd ban SATs tests being sent home for homework.
I'd expand testing to include every NC subject.
I'd anonymise results and wouldn't give any scores for individual children. Results couldn't be used beyond measuring the school.
Year 6 should just be a normal school year, school trips and plays, and a full curriculum including art and music, normal schools hours, no extra homework, and a short assessment week at the end of the year.