I went to a better-than-average public high school and then a top tier college and I spent lots of time with people who went to prep school. There are substantial differences between prep school vs. public school in a rich well educated area vs. public school in a poor area.

Prep schools are most certainly better (on average) at teaching students to be assertive, seek out opportunities, bend rules, etc. The coursework is harder and more engaging than even top-track classes at good public high schools. On average students get more individual attention from teachers.

Likewise, there is a wide gap between work difficulty and quantity and expected levels of preparation at top tier colleges vs. average colleges. At top tier colleges, the typical student is incredibly ambitious, and there is a categorical difference between the extent to which typical students seek leadership opportunities and pursue impressive goals. (This is not to diminish the work of typical students at average schools, many of whom come from difficult circumstances and must struggle to pay the bills, etc. And of course there are some amazingly ambitious and talented students even at mediocre schools who do amazing work.)

There are surely still many problems with even the best schools, but the difference is by no means limited to professional networks.