Yes, but there's literally no way to prove that a random number that you "generated" is actually random. The best idea would be to write people's names on balls, put them in a clear funnel thing, and blow them around or something like that on live TV, but for sortition, I'm not sure how that would scale. In the movie Contagion they did that for each day of the year someone could be born on, and that's how they distributed the vaccines. Not sure how that would scale to millions of balls and how you would audit that each person's name is on one ball and only one ball.
Yes, building a purely mechanical trustworthy source of randomness is hard.
However, you can look at how the lottery does it, perhaps?
For example, they don't write the players names on the balls. They use more indirection. (And lots of other cleverness.)
Also keep in mind that I suggest to use this mechanism for something like filling up a parliament. The German Bundestag has about 600 members. The UK has 650 MPs in their House of Commons.
I posit that getting into parliament is perhaps comparable to winning the lottery, ie something people might trust lottery equipment and procedures to handle.
Becoming the president of the US is a bigger deal than winning the lottery, so we can't naively expect lottery equipment and procedures to be above suspicion.
