While it probably varies between different fields, I haven't seen any evidence for that in the startup world.
There's obviously a certain level of "accident of birth", you have a much better chance if you were born in a developed country, one where you can go to university, etc. But if we standardise at say pool of CS graduates from good universities it's certainly a much more level playing field.
If we take what's probably the most important relationship in the startup world, that of co-founders, most co-founders meet each other through work, actively seeking through professional networks or university. There's very few co-founders who met through family connections.
Similarly for investors, investors will take a recommendation from one of their portfolio founders who've worked with someone much more seriously than that from a family friend who just knows someone socially.