Choosing people at random is not going to work well. Look at what happens with juries - lots of people find creative ways to get out of jury duty and that's for a way smaller commitment of time than a political term. The randomly chosen person won't want to be in government at all. Their life has just been totally up-ended through no fault of their own. Now no matter what they do, they're going to be wildly famous and wildly unpopular with huge segments of the population. All they wanted was to do their job and take care of their family, now they're getting harassed on the street by political enemies, having to go on TV, having to justify their decisions to journalists. Presumably some of the really unlucky ones will have to become ministers, but how do they even get picked?

No, the randomly chosen person has no reason to give a single damn about anything or anyone except themselves in that position. Heck half of them will be thinking: if I just refuse to turn up to work, maybe there'll be a recall and a new person will be randomly chosen.

Then take into account all the exceptions and justifiable reasons why people will be able to get out of this duty (e.g. because they're sick, because they're caregivers, because they're non-citizens, because they're mothers, because their company testifies that person is critical to their operations, etc). The ones you'll end up with are the ones who have nothing better to do. Expect the terminally unemployed to be over-represented.