I think you are confusing "eligible" and "get elected to the biggest, most visible positions".
Turns out, even in Athens, for the really big jobs they were kind of picky - dokimasia is an excellent filter to remove those you don't want. (Part of the process was questioning your "moral character", which is a pretty good proxy for "has an in with the powers that be").
And if we exclude high-visibility offices, there's a lot of people who can and do run in the US, for their local offices. School boards, city council members, commissioners, etc.
If we want to debate shortcomings of modern democracy, we might want to focus on the devaluation of local offices to the point of only focusing on national office.