I think if I were to send a bright child to a good, supportive, academic private school, I would expect them to achieve the best grades they were capable of achieving (barring disaster). I would also expect they'd gain a lot from a broader education that was not purely exam-focused, from being educated in an academically selective environment and from extra-curriculars.
If a very selective state school, I'd expect the same, academically.
Sending the same child to a very good, mixed ability state school, populated by enough bright, well-supported children to create a culture of achievement amongst that sub-group at least, I would hope they would achieve the same exam results and, if boosted by a bit of tactical tuition, might feel able to expect that.
Sending them to a less good, or very mixed ability state school, I would not feel able to expect that, even with tuition. The mixed quality of teaching, disruption to classes from teacher turnover, disruptive pupils, teachers having to focus desperately on securing passes for middle-range students, so not having time to focus on stretching the most able, or even to cover the whole syllabus in many cases... all of that would add up to a near impossibility of the best students achieving their potential, unless they adopted a completely independent, self-motivated approach to learning and/or were tutored to the extent that the parents were effectively running a parallel school and might as well home-educate or go private.
Picking up on the potential and motivation of that latter set of students is what I think contextual offers are and should be about.
The first two, even possibly three sets of bright children don't need to be as motivated as the fourth set, to achieve very good grades, because they're so well taught, supported and organised (partly by others). Strong motivation and self-organisation might boost them from very good to excellent. For the fourth set though, gaining good to very good grades is a feat demonstrating a degree of self-motivation the others have never been required, maybe even had the opportunity, to demonstrate.
So it seems to me that contextual offers are only about plucking the most capable and motivated students from the most difficult contexts. It doesn't capture the very capable but less self-motivated (who would do well in a selective, supportive school), or those actually defeated by their circumstances, who will already have fallen by the wayside. It's a very partial fix.
Real equality would look something much more like intervening to support and better educate all capable children (and all children), so that they can all achieve their potential, thus compete on an even playing field, when it comes to university entrance. That though, is a different discussion.