Today, sortition is commonly used to select prospective jurors in common law-based legal systems and is sometimes used in forming citizen groups with political advisory power (citizens' juries or citizens' assemblies).
People reading this are likely familiar with the use of sortition in jury selection. What is probably the single most well know thing about sitting on a jury?
It sucks. It's somehow boring and stressful at the same time, and the pay is very much token. Therefore, most people try to get out of it, with varying degrees of success. People have gone as far as not registering to vote in order to avoid jury duty. This is specifically why voter rolls are not used for that in many places.
The corollary to this is that juries aren't actually a random sample: people getting out of jury duty obviously causes selection effects. If the sample is not random, you lose a lot of the theoretical advantages of sortition. Unless we take strong steps to make sitting in a legislative body not suck, I see no reason it would be any better there.