No, they don’t always shine wherever they go. We gave relatives who believe this. Indeed one DC has done well but the other two barely do anything and get away with not doing homework and have gone backwards since primary Sats and definitely are squandering potential.
Some parents do think that paying somehow elevates the intelligence of the child. Of course it doesn’t. However many privately educated DC find themselves being taught with DC whose parents do believe in a good education. They all have the same goals and that’s very useful.
You should never compare results either unless the independent school, by some miracle, has an identical intake. It almost certainly won’t have the same SEN children and one assumes no PP children at all. Therefore look at added value but the private school might not publish this in any meaningful way.
Education at a private school should be broad and inspiring. It really isn’t just about results. It’s about ethos, music, sport, inspiring speakers, really good teachers, excellent facilities, tradition, fun and finding your tribe. If you go private just for results, I wouldn’t bother if it’s not an academic school. If the school gives you lots of clubs, sport, extras and inspiration, then it’s worth it.