First, people talking about the Athenian system - it was limited to "Free men" of Athens. The discrimination might be something you feel we could counter today - all men and women could be selected - but what about convicted felons? Would they have a right to participate? Today, one is President of the USA (yes, I know, many think that's a sham, but legally he is). What about gay people? 50 years ago they almost certainly wouldn't have been allowed in. 150 years ago black people wouldn't. What about today's political third rail populations? Would communists or socialists or anarchists be allowed in? What about trans? Or people not born in the USA?
There will always be a group some part of the spectrum wants to other and exclude, but today there is a democratic short circuit: an Austrian can be voted into being the Governor of California; a convicted felon can be voted into the position of President; a transgender woman can be voted into Congress by her Delaware constituents.
Balloting - almost like voting today - will get systematic tinkering. Voting isn't perfect. Gerrymandering in the US is out of control and a more neutral system like the UK's needs creating, and yeah, donations are a mess...
... but choosing people at random is never really random, and there is no easy way to short circuit the orthodoxy. Minorities will get better representation at first, but they'll always be minorities and if the majority wants rid, they'll be locked out forever, just the same way women and slaves were in Athens.