SATS are complently irrelevant as they measure school ability as opposed to pupils ability. Therefore I would leave it to teachers to decide what to do with them or how to revamp them. However I would strongly support tests that specifically measure a pupil's progress both in breadth and in depth, every six months in all years of the primary school. All should be school administered and blindly marked and then moderated. More importantly what matters in not the test but what you do with them. They are supposed to provide the basis for providing feedback, support and remedial (if needed) action
by the teacher to the pupil on the basis of the weaknesses as identified in the test. If this does not happen then any test is completely and totally meaningless. They can just take their GCSEs and A levels and be done with.
This is precisely what the frequent tests in the grammar school of my son do. He had one "blip" in a test in history and another in english. Follow up action which ensured coursework and classwork were properly redone and completed was immediate and he had to attend 2 clinic sessions to ensure that any gap in the particular topic have been dealt with. Similar action was taken following the test in English. The history topic that caused the trouble was on Blo-ody Mary! It is nice (?) to know he no longer has any gaps in it