@HungryGuy makes good points.
In addition, as bad as current democratic systems are, and they are pretty bad, no doubt, having “random” people selected (and how would they all be vetted as to Constitutional qualifications for serving? not to mention that the US Constitution, for one, would need major revision). Another argument is “residency”. In the US, people frequently enough have multiple domiciles – or none at all – so what would determine which district a person could be selected from? Aside from these administrative problems, the government now is simply too complex for most average (and let’s face it, “below-average”, citizens and intellects to comprehend). We’d have something far worse than we now have.
No, I think there are too many problems with that suggestion.
However, in talking to people around the US (and I expect the same holds true of other countries as well), people have a generally favorable opinion of “their guy” (even when they voted against him; at least now he’s “their guy”) just like they have great opinions of “their local school” even though “schools in general” are awful. Congress is awful in the same way.
What I’d like to see is a way for people around the country to vote “against” a single politician whom they consider to be the most antithetical to their interests. So voters from all over the USA could vote “against” a write-in candidate from another district, anywhere in the country, whom they would most like to see “gone”.
The one candidate who gets the most votes-against would not be seated in either the House of Representatives or the Senate, no matter how big he won in his home district. That district (the only one in the nation) would require a special election for another candidate, since it would be equally unfair to seat “whoever came in second” in the first election. I guess it would be fair for the federal government to share in the cost of that special election, too.