How is the current system transparent in this regard? How do you know where your MP/Congressperson got their information from (since we apparently agree that representatives are often not expert in the subject matter)?
I don't know, but at least I know it's them making the decision. I know they're corruptible, and I know their information can be manipulated and can be incomplete. It's transparent because everybody knows this, that's why we can complain about it. In some cases, we can call out lawmakers for it and get things changed. We know which ones tend to decide badly, and we can vote against them. It's an important balance in the system. If I have no control over who votes, and the whole system is made to look objective, that balance and that transparency is lost.
Ok, so your main beef with the lottery system is the anonymity of the randomly selected lawmakers? Well, if you don't know who they are, neither do the lobbyists. You still have Berlusconi/Putin-style influence-by-media, but that's not any different from what you have now.
My beef isn't with the anonymity, my beef is with the supposed objectiveness of the educators they use to give the randomly selected lawmakers the knowledge they need to decide. Those are the people to lobby, those are the people to buy, but they are hidden from the rest of the population. The only way to keep them honest would be to make every bit of information they give to the lawmakers public and freely available, so we can be sure the lawmakers aren't being manipulated. That runs into the same problems the delegate democracy has too: you can't make everything public in a state that still has foreign affairs.
My understanding is that only few representatives get close to state secrets, and they get them all from the executive branch (eg, the Intelligence Committee is relying on whatever the country's intelligence services deigns to tell them). Don't see why it should be different with randomly-selected lawmakers.