Yes, it's terrible your marriage has broken down, but don't ruin it for the rest of us because of the perceived shame at having to take your kids out of a posh school.
I’m pretty sure it was nothing to do with “shame” and everything to do with trying to minimise the disruption to his kids’ lives and ensure continuity of education to avoid any impact on their exam success. They were already having to cope with the trauma of divorce.
Having divorced when my kids were 10 and 13, this was also an issue I faced (keeping them in school, not claiming CEA). Thankfully I was well enough paid that I was able to keep them in their schools, whilst also paying alimony and child support! Very painful, financially, but worth it in the long run.
So, I sympathise but don’t condone.