The main problem with choosing representatives by lottery is that the average person just doesn't want to do the job. I know I wouldn't. It's about as appealing as having to do jury duty for three years.
The book proposes financial incentives as a way to address this reluctance, such as paying the person 1.5 times their current salary. But that seems like it will lead to representatives who are only doing it for the money, not because they care about doing a good job.
And then, of course, there's always a chance that you end up in a "Napoleon of Notting Hill" situation: https://en.wikipedia.org/wiki/The_Napoleon_of_Notting_Hill
Part of me wonders what would happen if a lottery were used not to select representatives from among the population at large, but from among the candidates who have won their parties' respective primaries. Of course, then you'd have to decide whether to use weighted selections (which would strongly favor the entrenched parties) or non-weighted (which would give a strong edge to minor parties).
But realistically, sensible term limits seem like they would help achieve a lot of what the proposed lottocracy system would achieve.