Random selection is necessary to create stable meritocratic institutions

One catch with any such system is that it effectively gives more power to people who are more motivated to actively participate in the process, which correlates with having stronger and more extreme political opinions. One could argue that it is only fair - everyone has the power to participate, after all, and if some people choose not to, they can't complain about the end result. But even so, an endless bitter fight between political extremes is unlikely to result in good governance (and I'm saying this as someone with fairly extreme political positions).