There have been no studies into GCSE attainment for incoming pupils with and without SAT data. In fact, no GCSE progress data is published (or even collected) for pupils without SAT data. As such, the detached quantitative evidence you're after cannot exist because the data for it is deliberately not collected. Anecdotal evidence is the only evidence that exists.
It's also why you have no non-anecdotal evidence for statements like:
many independent schools do not do SATS and when pupils transition to a secondary school they aren't held back as a result.
because no data on the progress of these pupils to GCSE is ever collected.
You are misunderstanding me. This isn't about whether teachers can identify whether or not pupils are underperforming. They certainly can do this in relation to how they believe that pupil should perform. It's about what, if anything, the secondary does in response to this underperforming.
One of the main ways that secondaries are held to account is by their progress 8 measure (this tracks the children from Y6 to Y11 and identifies whether they have made the expected level of progress, less than they should have or more than they should have). School leaders are incredibly mindful of this and factor it in to their decision making process, particularly when resources are limited.
To work through a practical example, if there are three children who are all currently on track for a 4 or a 5 in maths and the school has the ability to offer one of them additional support through mentoring, they will use KS2 data and progress 8 to decide which one gets it. For example:
If Pupil A is getting a 4 or 5 but only needs to get a 3 to hit a neutral progress 8 score, they are already 'overperforming' and won't be selected.
If Pupil B is getting a 4 or 5 and their target for a neutral progress 8 score is a 4 or 5, they will be selected unless...
Pupil C is getting a 4 or 5 but should be getting a 6, based on their KS2 results. In that case, Pupil C will be selected over Pupil B and Pupil A.
All three pupils would have benefitted from the mentoring but Pupil C gets it because, data-wise, they are a higher priority and it's a limited resource. Pupil C, as a result of the mentoring, manages to get that higher grade at GCSE.
We can add in Pupil D, who has no KS2 data and so will never contribute a progress 8 score (negative, neutral or positive) to the school's total. Pupils C, B and A would all get the mentoring before Pupil D because their progress affects the secondary school and Pupil D's doesn't. He is, statistically, irrelevant.