"They want a quick and easy way to decide among unknown people who to vote for."
Maybe so, but is that an option you'd really want to give people? That just makes the already-lazy even lazier.
"I wouldn't want to hire a plumber to be my surgeon."
That's not what including a wider range of professions is. It's asking a plumber to serve as an advisor of sorts to the surgeon that you also hired. When you're doing surgery, the plumber knows his limits, knows what he doesn't know, and doesn't interfere. On the contrary, when the surgeon needs his plumbing fixed, he has a plumber to help him out instead of doing so himself. When things fall in between, we turn to interest groups and lobbyists and things like that just as we do now. The difference is that instead of lawyers and businessmen that know how to complicate the English language and somewhat responsibly distribute money, we have lawyers, businessmen, scientists, teachers, anthropologists, police officers, doctors, preachers, or whatever to add some insight into whatever, and they split up into committee by profession.