Public politics, and private company politics are very similar, although private company politics are less open to scrutiny.
The issue with the lottery is the need to ensure that the candidates both want the role, and are capable of doing it.
The latter, who is the right person to say "X is unqualified because.. " (and the Peter Principle suggests that just because someone was good at a lower job, eventually they're going to be put into a job they are unqualified for)
The theory with the current style that the person who puts themselves forward most definitely desires to win the job, and, as they rise up through their party system, have some level of competence, as adjudged by the people they have convinced to put them forward as a candidate.
Further, the adversarial nature is supposed to then mean that that person's opponents can call out the reasons that that person isn't suitable for the job.
Unfortunately, this ends up being a muck raking exercise, and the complaints might not amount to anything more than innuendo, further, there's no guarantee that they will even be heard (the supporters will provide evidence that the opponents themselves are not qualified to make any criticism)
Unfortunately a lot of elections these days, US or otherwise, tend not to end up being "This candidate is awesome, let's vote them in", but, instead "the incumbent is terrible, get someone, anyone, to replace them" - in the US Biden was voted in because Trump 1.0 was deemed a failure, and then Trump 2.0 was voted in because Biden was deemed a failure. Right now the Democrats appear to be on the rise again because Trump 2.0 and the Republicans are being deemed a failure. This isn't to diminish the wins by some actually good candidates though (although how good they are remains to be seen, and is a matter of... opinion).