I went to a boarding school for years.
I have precisely ZERO connections. Please can we drop this myth of connections?
Perhaps Eton? Perhaps Radley?
None of the people I knew at my school went on to achieve anything remarkable including me. Connections might exist within the top elite but then they always have.
OP it’s the sport, drama and music that will be areas to consider.
And yet again now I have my kids at private school - after primary state school, but there are kids there that do nothing. No sport, drama or music. None of them are essential in life. It’s nice but things like a language, learning to code, learning to do well at public speaking, doing D of E or cadets, volunteering, learning to treat people with respect and kindness. These things are much more important in my opinion. They’ll help you get a job more easily too than saying “I’m grade 8 violin and I was in the rugby A team”.
Those latter accomplishments seem more to boost the school’s prowess rather boost your child’s chances of landing a good job. Don’t be mislead by glossy brochures.