Yes I'd take this part with a pinch of salt OP.
The UK national curriculum is designed the way it is so children access it in a spiral, planting seeds then revisiting them to nurture them until they grow into massive trees of understanding.
If you try to skip ahead or hothouse them, the children don't actually do the deep learning/proper retention/understanding of the underlying concepts required to apply that learning to the next thing. The pressure to always "be better" and get ahead of your peers causes children to burnout at different points along the way. Some do it around GCSEs, some do it in the sixth, and some do it during or after uni.
The NC isn't perfect but it's pretty damn good, it's designed to meet children at the developmental milestones they're at (such as stages of brain development) and decent private schools don't try to skip it or get ahead of it.
Anyone who knows anything about how children learn would never say a child is X years behind at primary, it's a meaningless statement.
This is why hothousing is a bad idea, and good schools (private, public or state) don't engage in it.
I'm not anti private schools at all (DH works in one), but from what you've said I'd advise you to go with caution OP and look through the marketing to what they are actually offering, as difficult as that can be. Some private schools are worse than state schools and they're still expensive.